Quelques aspects de la g©om©trie non commutative en - Science

Quelques aspects de la géométrie non commutativeen liaison avec la géométrie différentielle

Thierry Masson

LPT-Orsay (UMR 8627)Université Paris XI

Mémoire d’habilitation à diriger des recherches

présenté le 17 février 2009 devant le jury composé de

Daniel BENNEQUIN, Université Paris Diderot-Paris 7Michel DUBOIS-VIOLETTE, Université Paris XIMarc HENNEAUX, Université Libre de Bruxelles, rapporteurGiovanni LANDI, Université de Trieste, rapporteurClaude ROGER, Université Lyon I, rapporteurJean-Christophe WALLET, Université Paris XI

Table des matières

Introduction générale 5

1 Ideas and concepts of noncommutative geometry 151.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2 C∗-algebras for topologists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16General definitions and results – The Gelfand transform – Functional calculus

1.3 K-theory for beginners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23The topological K-theory – K-theory for C∗-algebras – Algebraic K-theory

1.4 Cyclic homology for (differential) geometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Differential calculi – Hochschild homology – Cyclic homology

1.5 The not-missing link : the Chern character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53The Chern character in ordinary differential geometry – Characteristic classes and Chern characterin noncommutative geometry – The Chern character from algebraic K-theory to periodic cyclichomology

1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2 SU(n)-principal fiber bundles and noncommutative geometry 652.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.2 A brief review of ordinary fiber bundle theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Principal and associated fiber bundles – Connections – Gauge transformations

2.3 Derivation-based noncommutative geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Derivation-based differential calculus – Noncommutative connections and their properties – Twoimportant examples

2.4 The endomorphism algebra of a vector bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79The algebra and its derivations – Ordinary connections

2.5 Noncommutative connections on A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Main properties – Decomposition of noncommutative connections on the module A –Yang-Mills-Higgs Lagrangian on the module A

2.6 Relations with the principal fiber bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86The algebra B – Ordinary vs. noncommutative connections – Splittings coming from connections

2.7 Cohomology and characteristic classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90The cohomology of ΩDer(A) – Characteristic classes and short exact sequences of Lie algebras

4 Table des matières

2.8 Invariant noncommutative connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Action of a Lie group on a principal fiber bundle – Invariant noncommutative connections

Bibliographie 101

Introduction générale

Pour rédiger ce mémoire d’habilitation, j’ai opéré deux choix :m sur le fond, j’ai opté pour ne faire apparaître qu’une partie de mes travaux de recherche menés

depuis plus de ans dans le domaine de la géométrie non commutative ;m sur la forme, j’ai repris, comme chapitres principaux de ce mémoire, deux articles de revue

que j’ai écrit respectivement en et ([Masson, a] et [Masson, c]), qui sont descomptes-rendus de conférences données lors de rencontres internationales.

Le but de cette introduction est de replacer dans leurs contextes respectifs ces deux revues, et deles compléter sommairement sur certains points.

La géométrie non commutativeLa géométrie non commutative a été conçue à la fois pour répondre à des besoins en mathéma-

tiques et pour permettre d’aborder certains problèmes de physique théorique.En mathématique, il s’agit de généraliser les outils de la géométrie ordinaire qui ont été déve-

loppés depuis plus d’un siècle : structures diérentiables, métriques, actions de groupes, brations,connexions. . . Ces constructions mathématiques sont désormais largement utilisées en physiquethéorique, et l’essentiel des théories modernes (le Modèle Standard des particules élémentaires, laRelativité Générale, la théorie quantique des champs) se fondent sur des propriétés mathématiquesnes élaborées dans ce contexte. C’est à A. Connes que l’on doit, en , d’avoir donné les premièresvoies concrètes de recherches dans le domaine de la géométrie non commutative, en dénissant et enétudiant la cohomologie cyclique [Connes, ]. Il montrait ainsi que la notion de calcul diérentielsur les variétés admet un équivalent non commutatif, au sens expliqué ci-dessous.

En physique, dès le milieu des années , les travaux sur les théories quantiques des champsont permis de faire émerger des notions devant faire cohabiter structures géométriques et structuresalgébriques : d’un côté les algèbres d’opérateurs, de l’autre les théories de jauge, c’est à dire la théoriedes connexions sur les brés. . . Cependant, la plus forte motivation reste encore aujourd’hui l’es-poir d’écrire une théorique quantique de la gravitation avec cette mathématique, puisque le principefondateur de la géométrie non commutative est de fusionner dans un même cadre conceptuel l’as-pect opératoriel de la mécanique quantique et l’aspect géométrique de la Relativité Générale (et desthéories de jauges).

L’idée maîtresse de la géométrie non commutative est d’abord de caractériser une classe d’es-paces « géométriques » bien particulière par un type d’algèbres de fonctions adapté, en munissantces algèbres d’outils algébriques appropriés. Par exemple, il est possible de caractériser un espacetopologique compact et séparé par son algèbre de fonctions continues bornées, et un espace mesu-rable par son algèbre des classes de fonctions mesurables bornées. Dans les cas favorables, ou biences outils algébriques n’utilisent pas explicitement la commutativité des algèbres de fonctions, oubien des outils plus algébriques équivalents existent, ce qui rend alors possible l’étude des algèbres

6 Introduction générale

non commutatives du même type à l’aide de ces outils algébriques, sans avoir à mentionner d’espacesous-jacent.

Cette démarche est largement encouragées par de nombreux résultats mathématiques : caracté-risations des algèbres de von Neumann commutatives, théorème de Gelfand-Naïmark sur les C∗-algèbres, théorème de Serre-Swan sur les brés, cohomologie cyclique de l’algèbre C∞(M), relationsentre opérateurs de Dirac et métriques riemanniennes, K-théorie des C∗-algèbres. . .

Le Chapitre est une revue non exhaustive à vocation pédagogique de ces considérations gé-nérales sur la géométrie non commutative. En particulier, l’eort pédagogique a été porté sur lesobjets mathématiques de nature géométrique qui admettent une généralisation non commutative,comme par exemple le caractère de Chern, pierre angulaire de nombreux résultats en géométrie noncommutative.

Ce chapitre ne constitue qu’une introduction à certaines structures algébriques et géométriquesqui fondent la géométrie non commutative. On trouvera dans mon ouvrage de pages, Une in-troduction aux (co)homologies, publié en aux éditions Hermann ([Masson, b]), de quoilargement compléter ce qui est exposé ici. En particulier, on trouvera dans les nombreux exercicesproposés des exemples concrets d’utilisation de ces outils dans la démarche de la géométrie non com-mutative. Pour des introductions plus complètes, je renvoie à [Connes, ], [Landi, ], [Madore,] et [Gracia-Bondía et al., ].

De plus, ce chapitre ne contient aucune contribution personnelle aux mathématiques de la géo-métrie non commutative sur lesquelles j’ai pu travailler par le passé : problèmes cohomologiquessur des algèbres associatives ([Dubois-Violette and Masson, a]), notion de sous variétés noncommutatives et de variétés quotient non commutatives dans le cadre de la géométrie non commu-tative basée sur les dérivations ([Masson, ]), étude de la notion d’opérateur diérentiel du pre-mier ordre sur un bimodule en géométrie non commutative ([Dubois-Violette andMasson, b]),structure du calcul diérentiel basé sur les dérivations de l’algèbre des matrices ([Masson, ],[Masson, b]). . .

Gravitation et géométrie non commutativeUne des motivations fortes de la géométrie non commutative est d’obtenir un cadre mathéma-

tique cohérent dans lequel il serait possible d’écrire une gravitation quantique. Bien que ce thème nesoit pas repris plus loin dans cemémoire, il me semble important dementionner et discuter quelquesuns des travaux menés dans cette direction.

Une des pistes explorées a été de généraliser le formalisme utilisé dans le cadre de la RelativitéGénérale d’Einstein à des situations opératorielles. Pour ce faire, une première étape incontournableconsiste d’abord à considérer les notions ordinaires de la géométrie riemannienne de façon plusalgébrique, et à envisager ensuite de se passer de la commutativité de l’algèbre des fonctions C∞.

J’ai mené par le passé, avec diérents collaborateurs, des travaux dans cette direction. Ils consis-taient à étudier une notion de connections linéaires non commutatives. Les connections linéaires sont,dans le formalisme de la Relativité Générale, l’objet mathématique derrière les symboles de Christof-fel. La dénition en elle-même des connections linéaires non commutatives ne pose pas de problème,car elle est déjà de nature très algébrique en géométrie ordinaire. On trouvera dans [Dubois-Violetteet al., ] et [Dubois-Violette et al., a] des considérations générales à propos de ces aspectsalgébriques.

Les géométries non commutatives sur lesquelles ces notions ont été testées sont très diverses : al-

. Même si cet ouvrage, comme son titre l’indique, n’est pas un livre sur la géométrie non commutative.


gèbre des matrices ([Madore et al., ]), plan quantique ([Dubois-Violette et al., b]), groupesquantiques ([Georgelin et al., ] et [Georgelin et al., ]). Depuis, d’autres auteurs ont exploréd’autres classes d’exemples, comme le plan quantique h-déformé ([Cho et al., ]), des géomé-tries de réseaux ([Dimakis and Mueller-Hoissen, ]), des géométries avec métriques pseudo-riemaniennes ([Dimakis and Mueller-Hoissen, ]). . .

De tous ces travaux, il ressort que les contraintes non commutatives sont tellement fortes quel’espace des connections linéaires se réduit très souvent à un espace de paramètres de dimension nie.Ceci est souvent relié au fait que le centre de ces algèbres est luimême très petit, un espace vectoriel dedimension nie. Cette situation est peu encourageante dans l’espoir de faire de la Relativité Généraleau « sens ordinaire ».

Une autre approche possible consiste à envisager que la quantication de la Relativité Généraleinduise une notion d’espace non commutatif, ayant pour paramètre de déformation la longueur dePlanck. Il ne s’agit donc pas de réécrire d’une façon ou d’une autre la Relativité Générale directement,mais d’en extraire d’éventuelles conséquences quantiques et de les encoder dans un espace-temps noncommutatif. Souvent, cet espace-temps non commutatif est étudié d’une point de vue de sa géométrie« riemannienne », ou bien il est le support de théories des champs dont on étudie les propriétés. Jerenvoie à [Madore, ] pour de plus amples renseignements sur ce domaine de recherche.

Enn, une autre voie de recherche pour écrire une théorie de la gravitation au moyen de la géo-métrie non commutative a été initiée et très largement explorée par A. Connes et A. Chamseddine(voir [Chamseddine and Connes, ] pour la première proposition, et [Chamseddine et al., ]pour la dernière version de ce modèle et les références bibliographiques). L’ambition de cette dé-marche, qui aboutit à un modèle assez complet dans le dernier article cité, est de reconstruire leModèle Standard de la physique des particules élémentaires couplé à la gravitation, à partir de prin-cipes issus d’idées de la géométrie non commutative.

Cette démarche ne cherche pas, contrairement aux précédentes, à envisager d’utiliser la géomé-trie non commutative pour écrire une «gravitation quantique», puisque lemodèle auquel elle aboutitest classique. Dans un soucis de comparaison avec les autres approchesmentionnés dans les chapitresde ce mémoire (sur les théories de Yang-Mills-Higgs non commutatives), il est utile de rappeler ici,en quelques lignes, en quoi consiste cette démarche.

La géométrie non commutative appliquée à la physique qu’A. Connes a développée est très dié-rente de celle qui sera expliquée au chapitre . En eet, la construction d’A. Connes repose de façonessentielle sur un triplet spectral, constitué d’une algèbre topologique, d’un espace de Hilbert sur le-quel cette algèbre se représente, et d’un opérateur de Dirac sur cet espace de Hilbert. Ces trois objetssatisfont à des relations de compatibilité, qui ne sont, ni plus, ni moins, que ce qu’il faut pour quel’algèbre des fonctions C∞ sur une variété compacte à spin, muni de l’espace de Hilbert obtenu parcomplétion des sections L d’un bré des spineurs et de l’opérateur de Dirac naturel, soit un tel tripletspectral. C’est le modèle commutatif de cette construction.

Cette notion de triplet spectral se transforme rapidement en un quintuplet spectral lorsqu’onlui ajoute une éventuelle Z-graduation et une notion de réalité. Ces deux concepts, formalisés pardeux opérateurs γ et J sur l’espace de Hilbert, trouvent leurs origines dans le Modèle Standard desparticules : γ est relié à la chiralité à travers l’opérateur usuel γ, et J se relie à la conjugaison decharge. Il ne s’agit pas ici d’exposer en détail ces constructions, de nombreux articles et ouvrages endonnent des exposés très précis ([Connes, ] ou [Gracia-Bondía et al., ] sont de bons pointsde départ).


Dans cette approche, l’opérateur de Dirac représente la métrique, et ses uctuations (non com-mutatives) s’identient à des champs de jauge. On peut, jusqu’à un certain point, comprendre cespropriétés par l’argumentaire heuristique suivant.

La métrique sur une variété diérentiable riemannienne est donnée par l’invariant de longueurds = gµνdxµdxν. Il est bien connu que l’action naturelle que l’on écrit dans le cadre de la théorie dela relativité, pour une particule libre (voir par exemple [Landau and Lifchitz, ]), est

S[γ] = ∫γ dτoù γ est un chemin paramétré sur la variété, et dτ =

√ds est ici le temps propre innitésimal le long

de ce chemin (dans une signature Minkowskienne). Donc S[γ] est le temps propre de l’objet le longde la trajectoire γ.

En mécanique quantique relativiste, on est amené à remplacer le formalisme des points et destrajectoires par le formalisme des fonctions d’ondes et de leurs équations diérentielles, grâce à la« procédure de quantication » pµ ↦ −i ∂

∂xµ . Dans l’espace des impulsions, l’invariant correspondantà ds estm = gµνpµpν, qui, quantié, donne lieu au Laplacien ∆ (généralisé, au sens où il peut s’agirdu d’Alembertien). Il est possible de reproduire la démarche qui conduit à l’action dans l’espace despositions en considérant un opérateur D tel que D = ∆, c’est à dire un opérateur de Dirac. Uneaction naturelle serait alors de la forme ∫ D, qui n’a de sens que dans le cadre de la théorie spectraled’une certaine classe d’opérateurs. L’action spectrale proposée par A. Connes et A. Chamseddine estde la forme

S[D] = ∫ f(D)

où f est une fonction dénie sur le spectre de D.Imaginons maintenant que l’on permette à D d’avoir des uctuations dans un espace susam-

ment « grand », par exemple de telle façon que les couplages minimaux, qui consistent à « faire uc-tuer » les pµ sous la forme pµ + Aµ (Aµ étant un champs de Yang-Mills), soient possibles. L’actionécrite ci-dessus sera alors une action contenant les champs gµν et Aµ. La structure de cette actionimpose que S[D] produise des invariants mathématiques à partir de ces champs. Aussi, le travail deA. Chamseddine, A. Connes et M. Marcolli, dans [Chamseddine et al., ], a consisté à permettreles bonnes uctuations de D (en choisissant, en particulier, l’algèbre et l’espace de Hilbert du tripletspectral) de telle façon que cette action reproduise une version du Modèle Standard des particulesélémentaires (il s’agit d’unmodèle inspiré desmodèles see-saw, contenant unmélange des neutrinos).

Il est bien connu que cette démarche soulève de nombreux problèmes. Techniquement, l’opéra-teur de Dirac qu’il est possible de considérer dans cette approche est à résolvante compacte, ce quiexclu de fait les opérateurs de Dirac de la physique sur espace-temps de Minkowski. Seuls les caseuclidiens peuvent donner lieu à de telles modélisations, ce qui rend l’intérêt physique de cette dé-marche, dans son formalisme actuel, assez peu encourageante. Certains auteurs ont commencé àexplorer la possibilité d’un opérateur non elliptique. Pour cela, il a été suggéré d’amplier encore letri(quintu)plet spectral en lui adjoignant un opérateur supplémentaire, un peu à la manière où unespace de Krein est un espace de Hilbert muni d’un opérateur métrique qui relie le produit scalaireindéni au produit scalaire déni positif. . .

Une autre diculté de la construction d’A. Connes est qu’elle requiert une certaine forme de

. Même si de trop nombreux physiciens pensent qu’une « simple » rotation deWick surait à basculer d’une signa-ture à une autre, ce que lesmathématiques ne semblent pas étayer : combien de théorèmes valables dans le cas elliptiquesle sont aussi dans le cas hyperbolique ?


compacité pour l’« espace » non commutatif. Dans [Gayral et al., ], la dénition d’un tripletspectral non compact a été proposée, et l’exemple d’une algèbre de Moyal (il y a de nombreusesdénitions possibles de l’« algèbre de Moyal ») y est donné. Cette approche requiert encore une foisd’élargir le triplet, en ajoutant une algèbre topologique non unitale. . .

Je reviendrai plus loin sur les points plus positifs de cette approche. Il ressort cependant de ce(très) bref tour d’horizon que la « gravitation quantique » est pour l’instant encore à un stade d’in-vestigation relativement préliminaire dans le cadre de la géométrie non commutative.

Relations entre la géométrie non commutative et la géométrie ordinaireLa géométrie non commutative se veut une généralisation de la géométrie diérentielle ordi-

naire. Aussi, il n’est pas surprenant de s’attendre à ce qu’elle permette de reconsidérer les objets de lagéométrie ordinaire dans son propre langage. Il en est ainsi, par exemple, de la notion de feuilletage :A Connes a montré qu’on pouvait associer et étudier une certaine C∗-algèbre à un feuilletage, même(et surtout !) singulier. C’est l’un des succès majeurs de la géométrie non commutative d’avoir puprendre en considération des situations singulières de la géométrie ordinaire qu’il était très dicile,voire impossible, à manœuvrer dans le formalisme géométrique usuel.

En collaboration avec M. Dubois-Violette d’abord, puis avec E. Sérié ensuite, j’ai exploré dansdiérents articles une autre situation où la géométrie non commutative rejoint et jette un regardnouveau sur la géométrie ordinaire. La publication [Masson, c] est une revue de l’ensemble dece qui a été considéré jusqu’à présent sur ce sujet et constitue le Chapitre de ce mémoire. Je donneun bref aperçu, dans ce qui suit, des motivations derrière ce travail et des résultats essentiels obtenus.

Le point de départ des recherches publiées dans [Dubois-Violette and Masson, ] consistaità généraliser des travaux antérieurs sur la géométrie non commutative basées sur les dérivationsde l’algèbre des fonctions à valeurs matricielles. Cette algèbre, étudiée par M. Dubois-Violette, R.Kerner et J. Madore, avait montré que la partie purement non commutative (l’algèbre des matrices)renfermait, du point de vue des théories de jauge non commutative, des degrés de liberté assimilables,au moins dans des modèles simples, à des champs de Higgs.

La généralisation que nous avons considérée dans [Dubois-Violette andMasson, ] repose surla constatation que cette algèbre de fonctions à valeurs matricielles, isomorphe à C∞(M)⊗Mn(C)(M est une variété diérentiable paracompacte), n’est rien d’autre que l’algèbre des endomorphismesdu bré vectorielM ×Cn. Dans cette situation, ce bré est trivial. Nous avons donc amorcé l’étudede la géométrie non commutative basée sur les dérivations de l’algèbre des endomorphismes d’unbré vectoriel orientable E de bre Cn, qu’on notera de façon générale A par la suite. Le groupe destructure d’un tel bré pouvant être réduit à SU(n), nous avons souvent appelé cette algèbre l’algèbredes endomorphismes d’un bré SU(n).

Dans l’article [Dubois-Violette andMasson, ], nous avons montré que l’aspect non trivial dubré E modiait en substance certains des résultats obtenus pour l’algèbre des fonctions à valeursdans les matrices. Par exemple, et ceci joue une rôle crucial par la suite, l’algèbre de Lie des dériva-tions de cette dernière algèbre se scinde, en tant qu’algèbre de Lie et module sur le centre, en unepartie purement non commutative (liée aux dérivations intérieures de l’algèbre des matrices) et unepartie géométrique (l’algèbre de Lie des champs de vecteurs surM). Cette décomposition canoniquepermet de considérablement simplier l’analyse de cette géométrie. Au contraire, dans le cas de l’al-gèbre associée à un bré E non trivial, cette décomposition n’a plus lieu. Cependant, nous avonsmontré qu’une connexion ordinaire sur E permet de scinder l’algèbre de Lie des dérivations en tantque module sur le centre, mais pas en tant qu’algèbre de Lie, en une composante non commutative(le sous espace des dérivations intérieures de l’algèbre) et une composante « commutative » (l’algèbre


de Lie des champs de vecteurs surM). L’obstruction à la possibilité de scinder cette algèbre de Lieen tant qu’algèbre de Lie est exactement mesurée par la courbure de la connexion choisie.

Le point essentiel de cette publication est que l’espace (ane) des connexions ordinaires sur Eest un sous espace de l’espace (vectoriel) des connexions non commutatives de cette algèbre desendomorphismes (voir le ¿éorème ..). Les degrés de liberté ainsi disponibles en sus de ceuxd’une connexion ordinaire sont, comme dans le cas trivial, assimilables à des champs de Higgs.

D’autre part, nous avons relié cette géométrie à l’algébroïde de Lie d’Atiyah surM.Dans la publication [Masson, ], j’ai poursuivi l’étude de cette algèbre, en montrant en quoi

elle généralisait convenablement, sur de nombreux points, la géométrie ordinaire du bré principalP sous-jacent au bré E . La possibilité de relier entre elles ces deux géométries repose sur l’utilisationde mes travaux antérieurs sur les sous variétés non commutatives et les variétés quotients non com-mutative dans le cadre des calculs diérentiels basés sur les dérivations ([Masson, ]). En eet, onpeut considérer que les deux géométries mentionnées, la géométrie ordinaire de P et la géométrienon commutative de l’algèbreA, sont des géométries de « variétés quotient non commutatives » de lagéométrie non commutative de l’algèbreB des fonctions surP à valeurs dansMn(C). Cette dernièregéométrie est triviale au sens du bré vectoriel sous-jacent (voir la Section .). J’ai ainsi pu montrerque la réécriture des connexions ordinaires, comme connexions non commutatives dans le cadre del’algèbre A, complétait parfaitement le schéma géométrique habituel des connexions qui consistaitjusqu’à présent à les dénir comme formes sur P (équivariantes et horizontales, à valeurs dans unealgèbre de Lie) ou comme une famille de formes locales surM (à valeurs dans une algèbre de Lieet satisfaisant à des relations de recollement non homogènes), en dénissant une telle connexioncomme forme non commutative globale surM « à valeurs dans A ». Au niveau des courbures, cettedernière caractérisation existait auparavant (-formes surM à valeurs dans un bré associé à P).J’ai donc montré que la forme de connexion pouvait elle aussi se dénir à ce niveau intermédiaire(entre une notion locale surM et une notion globale sur P) en ayant recourt à des structures noncommutatives (voir la Remarque ..).

Dans cette publication [Masson, ], j’ai aussi considéré la structure de l’espace de cohomologiedes formes diérentielles non commutatives, et j’ai pu démontrer une généralisation non commu-tative du théorème de Leray sur la cohomologie d’un bré principal (¿éorème ..).

En collaboration avec mon étudiant de thèse, E. Sérié, nous avons déni et étudié dans [Massonand Sérié, ] la notion de connexions non commutatives invariantes sur cette géométrie noncommutative. Les connexions (ordinaires) invariantes sous l’action d’un groupe de Lie jouent unrôle important en géométrie ordinaire et en physique des théories de jauge non abéliennes. En eet,elles permettent bien souvent de trouver des solutions explicites des équations du mouvement enprésence d’un principe de symétrie réduisant considérablement les degrés de liberté.

La dénition que nous avons prises pour les connexions non commutatives invariantes sousl’action d’un groupe de Lie est une généralisation de la notion habituelle, au sens où l’espace desconnexions non commutatives contient l’espace (ane) des connexions ordinaires, et que les deuxnotions de connexions invariantes coïncident sur ce sous espace.

Nous avons en outre caractérisé l’espace des connexions non commutatives invariantes et donnéplusieurs exemples, en particulier une généralisation de l’ansatz deWitten. La démarche a consisté àcaractériser les connexions non commutatives invariantes comme des objets au niveau de la grandealgèbreB. L’espace des connexions non commutatives invariantes est alors constitué de deux parties :une algèbre et son calcul diérentiel, et un module sur cette algèbre. On retrouve ainsi la décompo-sition usuelle en une partie « connexion réduite » et une autre partie « champs scalaire ». Le langagegéométrico-algébrique permet de considérer et manipuler ces objets de façon beaucoup plus aisée


que dans le cas de la géométrie ordinaire (voir la Section .).Le Chapitre est une revue complète de tous ces travaux. En particulier, j’y insiste sur le lien

entre la géométrie ordinaire du bré P et la géométrie non commutative de l’algèbre A. Dans cetexposé, on trouvera aussi des résultats (non publiés par ailleurs) sur la façon de dénir les classescaractéristiques de E en terme de l’algèbre de Lie des dérivations de A. En eet, dans cette revue,je montre que l’obstruction au fait que cette algèbre de Lie se scinde en tant qu’algèbre de Liecontient les informations permettant de calculer les classes caractéristiques de E . Cette démarcherepose sur la notion de classes de cohomologie associées à une suite exacte courte d’algèbres de Lie,dénie et étudiée par Lecomte. On ne trouvera pas les démonstrations originales dans cette revue,et je renvoie aux articles originaux pour les détails.

On peut résumer brièvement ce qui a été appris sur cette géométrie non commutative, en di-sant qu’elle constitue un outil idéal pour le physicien désireux de considérer des modèles de typeYang-Mills-Higgs. En eet, la géométrie diérentielle des théories de jauge sur des brés SU(n) estune sous géométrie d’une géométrie non commutative de l’algèbre des endomorphismes de E . Cettecaractérisation des théories de jauge permet de mieux comprendre l’origine et la place des champsde Higgs vis-à-vis de la géométrie ordinaire, puisqu’ils s’interprètent, de façon tout à fait naturelle,comme les degrés de liberté dans les directions purement non commutatives de la géométrie de cettealgèbre d’endomorphismes. Ils s’agit donc d’une partie non visible à travers la géométrie usuelle. Ilfaut noter que dans le Modèle Standard élaboré par A. Connes évoqué plus haut, les champs deHiggs sont aussi les composantes des connexions dans des directions purement non commutatives(la « géométrie nie » de cemodèle). Ceci semble à la fois répondre à la question physique de l’originedes Higgs, et à la question mathématique du statut exact de ces champs scalaires dans les modèlesde jauge à brisures spontanées de symétrie, qui jusqu’à présent n’avaient pas été identiés mathéma-tiquement comme le sont aujourd’hui les champs de jauge (en tant que connexions).

La géométrie non commutative et les théories des champsCeci nous amène directement à l’un des thèmes de recherches les plus développés dans le do-

maine de la géométrie non commutative. Il concerne les théories de jauge non commutatives. Rap-pelons que pour dénir une telle théorie, c’est à dire dénir des connexions non commutatives, ilfaut trois ingrédients :

À une algèbre associative A, souvent unitale pour des raisons pratiques ;

Á un calcul diérentiel qui jouera le rôle de « géométrie diérentielle non commutative », sanslequel la notion de formes diérentielles n’a pas de sens ;

Â un module à droite sur l’algèbre A, qui sert de support à l’action de la connexion non com-mutative (dans les modèles de physique des particules, il correspond à l’espace des champs dematière).Bien souvent, on simplie le problème, tout en conservant une certaine généralité, en prenant

pour module sur A l’algèbre A elle-même, considérée comme module à droite sur elle-même.C’est par exemple avec cette démarche qu’avec mes collaborateurs j’ai étudié les connexions or-

dinaires comme connexions non commutatives sur l’algèbre des endomorphismes d’un bré SU(n)(voir Section .).

Une autre direction possible de recherche repose sur la constatation que les équivalents noncommutatifs des théories de jauge abéliennes sont des théories de jauge non commutatives dont lesdegrés de libertés s’interprètent comme des champs de jauge non abéliens ! Par exemple, en prenantl’algèbre A comme module sur elle-même, la structure non commutative de A correspond, en un


certain sens, au groupe de jauge : dans le cas A = C∞(M) ⊗ Mn(C), on trouve pour groupe dejauge C∞(M) ⊗ SU(n), ce qui rapproche la théorie de jauge (non commutative) considérée d’unethéorie de jauge de type Yang-Mills avec comme groupe de structure SU(n). Or, prendre l’algèbreelle-même dans le cadre commutatif (A = C∞(M)) revient à considérer une théorie abélienne detype Maxwell !

C’est dans cet esprit qu’en collaboration avec R. Kerner et notre étudiant de thèse E. Sérié, nousavons introduit des théories de type Born-Infeld non abéliennes. En eet, le caractère non abéliencorrespond au caractère non commutatif de l’algèbre A = C∞(M) ⊗ Mn(C), et nous avons alorsgénéralisé, dans une démarche purement non commutative, l’équivalent de l’action (abélienne) deBorn-Infeld. Ces travaux ont donné lieu à plusieurs publications. Dans [Kerner et al., ], en guisedemise en place de cette démarche, nous avons explicitement calculé le lagrangien dans le cas SU().Nous avons étudié (numériquement) les solutions statiques à symétrie sphérique de cette actionde Born-Infeld. Nous avons exhibé une famille à un paramètre de solutions d’énergie nie, dontnous avons décrit les propriétés essentielles. Dans [Kerner et al., ], nous avons réduit le typede lagrangien considéré précédemment à un ansatz ne faisant intervenir qu’un seul champ scalaire.L’action obtenue est de type Dirac-Born-Infeld. Nous avons étudié des solutions de cette action, dansdes cas simples à symétrie sphérique. Il en ressort que ces solutions ne sont pas stables.

Dans [Cagnache et al., ], en collaboration avec E. Cagnache et J.-C.Wallet, nous avons appli-qué la technologie des calculs diérentiels basés sur les dérivations à l’algèbre deMoyal, an d’étudierles conséquences possibles, au niveau des théories de jauge, du choix de ce calcul. En eet, l’algèbrede Moyal, qu’on peut considérer sur certains points comme une algèbre généralisant l’algèbre desmatrices Mn(C) largement évoquées ci-dessus, n’admet que des dérivations intérieures. Or, le cal-cul diérentiel basé sur les dérivations qui a été utilisé jusqu’à présent dans ce cadre, n’utilise pourespace de dérivations qu’un espace de dimension deux (pour simplier, on ne traite dans ce résuméque l’algèbre de Moyal sur R) : les « dérivées » dans les directions usuelles de R. Comme l’algèbrede Lie complète des dérivations est de dimension innie, il semble très restrictif de ne considérerque ces deux directions, tout en gardant à l’esprit qu’en vue de construire des théories de jauge, il estraisonnable et souhaitable de se restreindre à une sous algèbre de Lie de dimension nie. Aussi, nousavons montré qu’un choix tout aussi naturel serait de considérer une algèbre de Lie de dimension ,où, aux deux dérivations précédentes, on ajoute les directions du groupe des symplectomorphismes.Cette algèbre de Lie est la plus grande sous algèbre de Lie des dérivations de l’espace de Moyal pourlaquelle les dérivations sont aussi des champs de vecteurs usuels sur les fonctions ordinaires (dontcertaines, comme par exemples les fonctions polynomiales, sont dans l’algèbre deMoyal considérée).Nous avons étudié des théories de jauge dans cette situation, et montré que les degrés de liberté sup-plémentaires introduits sur les champs de jauge dans ces trois nouvelles directions pouvaient jouerle rôle de champs de Higgs.

La géométrie non commutative face aux « théories unificatrices »

Je voudrais terminer cette introduction en donnantmon point de vue sur la place de la géométrienon commutative dans le grand programme de recherche qui mobilise les esprits, assez vainementmalheureusement, depuis plus d’une cinquantaine d’années maintenant, et qui consiste à élaborerune théorie dans laquelle les interactions électrofaibles, fortes et gravitationnelles trouveraient leurplace d’une façon uniée.

Contrairement à d’autres approches du type « unicatrice » actuellement explorées, comme par


exemple la théorie des cordes ou la théorie des boucles quantiques , la géométrie non commutativene se place pas au niveau d’un modèle particulier dans une cadre conceptuel déjà pré-déni (les mé-thodes de quantication usuelles, la théorie quantique des champs et sa théorie des perturbations. . .).

Dans les faits et les articles publiés, cette assertion peut largement sembler inexacte, puisque bonnombre de chercheurs en géométrie non commutative n’adhèrent pas à ce point de vue. Pourtant,il me semble que cette conception soit la seule défendable lorsqu’on travaille avec cette nouvelleapproche : elle cherche en eet, avant tout, à puiser aux sources mêmes des théories an de dénicherce qui les rassemble, dans le but de les réconcilier (on connaît déjà depuis bien longtemps ce quiles divise !). Malheureusement, une large part des travaux se contente de reproduire et de répéterdes recettes bien trop classiques face aux enjeux que lancent cette unication, et dans la plupart dutemps éloignées de la philosophie même qui motive la géométrie non commutative. Cette dernièrecherche avant tout à fournir un nouveau cadre dans lequel penser et écrire une nouvelle physique,susceptible d’accueillir à la fois les aspects les plus pertinents de la théorie quantique des champs, etles caractéristiques les plus utiles des théories de nature géométriques. Par conséquent, les « recettes »habituelles doivent aussi y être repensées profondément.

La géométrie non commutative n’a pas encore atteint cet objectif, car les mathématiques qui lasous-tendent sont encore largement en gestation, et les physiciens ne se sont pas encore appropriéscette nouvelle démarche. Il n’est donc pas possible, à l’heure actuelle, de la comparer à d’autresapproches unicatrices.

Néanmoins, la possibilité de reformuler le Modèle Standard des particules en utilisant des outilsalgébriques nouveaux est un indéniable et très encourageant succès de la « reconstruction » eectuéeparA.Connes et ses collaborateurs. En eet, l’élaboration desmodèles de particules repose, depuis lesannées , sur l’utilisation d’espaces de représentations de groupes, et tout l’art du «model builder »,grande spécialité de ces dernières années, transmise de chercheurs aux thésards dans la bonnevieille tradition du compagnonnage, se révèle dans les choix adéquats des groupes de symétrie etdes espaces de représentation. LeModèle Standard est la grande victoire de cet Art.Mais depuis la ndes années , cette démarche semble s’essouer, non pas dans la littérature, mais dans ses succès,aussi bien pour dépasser le Modèle Standard (qui a bien besoin d’une cure de jouvence depuis larécente conrmation que les neutrinos sont massifs par exemple), que dans sa capacité à apporterdes perspectives nouvelles grâce auxquelles des théories unicatrices sont concevables.

Aussi, l’approche non commutative pourrait être un nouveau soue dans ce contexte. Dans tousles modèles évoqués ci-dessus, aussi bien celui très « réaliste » construit par A. Connes et ses col-laborateurs que les modèles plus bruts issus des considérations du Chapitre , la notion de groupede structure disparaît : elle s’eace derrière celle d’algèbre associative (la non abélianité des groupesse transformant en non commutativité des algèbres. . .), et les représentations sont remplacées pardes modules, pour lesquels l’irréductibilité est une contrainte plus forte. Enn, la non abélianité/noncommutativité apporte avec elle, de façon structurellement inévitable, des champs supplémentairesaux champs de jauge « ordinaires », interprétables comme des champs de Higgs dans les bons cas.

La démarche d’A. Connes devrait être suivie avec beaucoup d’attention par les physiciens des

. Bien que la gravitation quantique à boucles ne se revendique aucunement « unicatrice », elle présuppose quela gravitation puisse être quantiée avec des méthodes très semblables à celles utilisées pour manoeuvrer les autresinteractions, ce qui, de ce point de vue, est déjà un principe d’unication.

. Il faudrait d’ailleurs pour cela que nombre d’entre eux acceptent un changement de paradigme, ce qui n’est pasnécessairement le plus facile, compte-tenudes phénomènes demode oudes attachements psychologiques parfois rigidesà certaines démarches. . .

. Il est étonnant de voir combien l’apprentissage de la science à travers l’itinéraire des « post-docs » ressemble à ladémarche du compagnonnage.


particules. En eet, la construction même de sa géométrie non commutative est intimement liée,comme il l’avoue lui-même, aux structures géométrico-algébriques sous-jacentes au Modèle Stan-dard, qu’elles aient été introduites en toute conscience par les physiciens, ou qu’elles soient le résul-tat, au contraire, de contraintes fortes et inévitables face aux données expérimentales. Ainsi, les deuxopérateurs mentionnés auparavant, la graduation et la réalité, sont directement issus de ce chemine-ment. Ils permettent de mieux comprendre la structure mathématique globale que semble avoir leModèle Standard (si le modèle de Chamseddine-Connes-Marcolli est « relativement » correct).

D’autre part, il est admis que la «gravitation quantique» est un granddéd’unpoint de vue tech-nique, en particulier parce que nombre de méthodes pertinentes dans d’autres contextes de quanti-cation semblent inecientes dans le traitement de la gravitation. Cependant, il serait temps d’ad-mettre que le dé à relever est aussi et surtout de nature plus conceptuel. Je renvoie à l’excellente revuedeC. Isham [Isham, ] pour un exposé très complet et argumenté de ces questions fondamentales,qui ne semblent pas trop embarrasser les tenants d’approches « classiques », mais auxquelles il faudrapourtant bien apporter des réponses satisfaisantes avant même de vouloir s’attaquer aux problèmestechniques « concrets ». Ainsi, le statut de l’espace-temps est l’un des problèmes les plus profonds etpertinents qu’on ait à traiter et à résoudre. Comme le rappelle C. Isham, toutes les tentatives actuelle-ment mises en œuvre pour élaborer une théorie quantique de la gravitation (l’approche euclidienne,les cordes ou les boucles quantiques) présupposent une structure spatio-temporelle très « classique »sous forme d’une variété diérentiable on ne peut plus ordinaire. . . Face à ce type de dicultés, lagéométrie non commutative me semble beaucoup mieux armée car elle ouvre des perspectives in-concevables par ailleurs. En eet, comme je l’ai rappelé dans cette introduction, et comme l’illustre leChapitre , c’est le concept même de variété diérentiable que la géométrie non commutative se pro-pose, entre autres, de généraliser (avec succès !), sans qu’il soit nécessaire d’avoir recours aux notionsusuelles et embarrassantes (dans le contexte quantique) de points et de trajectoires. C’est pourquoije renouvelle mon espoir que la géométrie non commutative, après avoir enrichi considérablementles mathématiques en jetant des regards nouveaux et originaux sur bien des considérations, inspireenn, un jour, des idées fondamentalement nouvelles en physique.

Aussi, la piste géométrico-algébrique empruntée actuellement par la géométrie non commuta-tive, qui ne s’intéresse pour l’instant qu’aux théories de jauge, n’a pas encore rejoint la véritable na-lité qu’elle se propose d’atteindre en physique : permettre de penser une théorie (avant même de laconstruire. . .) éclairant de façon élégante et pertinente les deux aspects antagonistes de la Nature : lequantique et le géométrique. . .

. L’usage des guillemets me permet d’avoir à éviter de préciser d’avantage de quoi il s’agit précisément. . .

1

Ideas and concepts ofnoncommutative geometry

1.1 Introduction

Once upon a time, in a perfect land, the idea of point was conceived.¿is was a beautiful concept, full of potentiality, especially in Natural Science: how easy is it to

say where objects are when one has introduced such a precise denition of localization! How easyis it to describe the kinematics of bodies when one assigns to them a point at each time. . .Well, atleast if time is there too! And then laws were found for the interactions of moving bodies, and thenpredictions were formulated: Pluto, the former planet, was where it was calculated to be! Better:generalized geometries where conceived, in which parallels can meet. And you know it: physicists(one of them at least!) where fool enough to show us how useful these geometries can be to describegravitation.

But nature seems o en more subtle than human mind. And the dream ceased when quantummechanics entered the game. We are no more allowed to say where an electron is exactly locatedon its “orbit” around the proton in the hydrogen atom. What is the photon trajectory in the Young’sdouble slit experiment? It is forbidden to know! Knowing destroys the diraction pattern on thetarget screen.

¿emain feature of quantummechanics which exposes us to this annoying situation is the non-commutativity of observables.

How can we accommodate this? Well, one of the reasonable answers can be found in mathemat-ics. Surprisingly enough,mathematicians discovered, not so long ago, that we can speak about spaceswithout even mentioning them. ¿e trick is to use algebraic objects, and the surprise is that spaces(some of them at least, miracles are not the prerogative of mathematicians!) can be reconstructedfrom them. In the language of mathematics, one has an equivalence of categories. . .

¿e algebraic objects we need to deal with are associative algebras, not only with their friendlyproduct but also with other structures, like involutions and norms. ¿ere, quantum mechanics isat home: observables are special operators on a Hilbert space, so that they live in such an algebra!Where are the “points” which were mentioned? Take a normal operator (it commutes with its ad-joint) in such an operator algebra, consider the smallest subalgebra it generates. ¿is subalgebra is acommutative algebra, which can be shown to be the algebra of continuous functions on the spectrumof this normal operator. Associate to this element as many other normal operators as you can nd,

16 Chapter 1 – Ideas and concepts of noncommutative geometry

on the condition that they commute among themselves, and you get another algebra of continuousfunctions on a topological space. Yes, we get it: a topological space from pure algebraic objects!

¿is is one of the main results behind noncommutative geometry. ¿e idea is the following: ifcommutative algebras are ordinary topological spaces (in the category of C∗-algebra to be precise),what are the noncommutative ones? How can we study them using the machinery that we are usedto manipulate the topological spaces? Is there somewhere another category of algebras in whichcommutative algebras are dierential functions on a dierential manifold? If not (for the moment,it is no!), can we manipulate them with some kind of dierential structures?

I hope to show you in the following that these questions make sense, and that some answers canbe formulated. In section . we introduce C∗-algebras, and we make the precise statement aboutcommutative C∗-algebras. In section ., we show that one of the machineries developed on topo-logical spaces can be used on their noncommutative counterparts, the C∗-algebras. In section .,cyclic homology is shown to be a good candidate to fool us enough into thinking that wemanipulatedierential structures on algebras. Section . is devoted to the Chern character, an object whichcan convince the more commutative geometer that noncommutative geometry does not only makesense, but also is one of the most beautiful developments in modern mathematics.

1.2 C∗-algebras for topologists

In this section, we will explore some aspects of the theory of C∗-algebras. ¿emain result, we wouldlike to explain, is the theorem by Gelfand and Neumark about commutative C∗-algebras.

1.2.1 General definitions and resultsIn order to be concise, only algebras over the eld C will be considered.

Denition .. (Involutive, Banach and C∗ algebras)An involutive algebra A is an associative algebra equipped with map a↦ a∗ such that

a∗∗ = a (a + b)∗ = a∗ + b∗ (λa)∗ = λa∗ (ab)∗ = b∗a∗

for any a,b ∈ A and λ ∈ C, and where λ denotes the ordinary conjugation on the complex numbers.A Banach algebra A is an associative algebra equipped with a norm ∥ ⋅ ∥ ∶ A → R+ such that the

topological space A is complete for this norm and such that

∥ab∥ ≤ ∥a∥ ∥b∥

If the algebra is unital, with unit denoted by 1, then it is also required that ∥1∥ = .AC∗-algebra is an involutive and aBanach algebraA such that the norm satises theC∗-condition

∥a∗a∥ = ∥a∥ (..)⧫

AC∗-algebra is then a normed complete algebra equippedwith an involution and a compatibilitycondition between the norm and the involution. One can show that thisC∗-condition (..) impliesthat ∥a∥ ≤ ∥a∗∥ ≤ ∥a∗∗∥ = ∥a∥. ¿e adjoint is then an isometry in any C∗-algebra.

1.2 C∗-algebras for topologists 17

Denition .. (self-adjoint, normal, unitary and positive elements)An element a in aC∗-algebraA is self-adjoint if a∗ = a, normal if a∗a = aa∗, unitary if a∗a = aa∗ = 1when A is unital, and positive if it is of the form a = b∗b for some b ∈ A. ⧫

Self-adjoint and unitary elements are obviously normal, and positive elements are obviously self-adjoint.

Example .. (¿e algebra of matrices)LetMn(C) be the algebra of n×nmatrices overC. ¿is is an involutive algebra for the adjoint. ¿isalgebra is complete for the three equivalent norms

∥a∥max =max∣ai j∣ / i, j= , . . . ,n max norm∥a∥ = sup∥av∥ / v ∈ Cn, ∥v∥ ≤ operator norm

∥a∥∑ =∑i,j

∣ai j∣ sum norm

which are related by the inequalities

∥a∥max ≤ ∥a∥ ≤ ∥a∥∑ ≤ n∥a∥max

¿e algebraMn(C) is then a Banach algebra for any of these norms. Only the operator norm denesonMn(C) a C∗-algebraic structure. ⧫

Example .. (¿e algebra of bounded linear operators)Let H be a separable Hilbert space, and B(H) = B the algebra of bounded linear operators on H.Equipped with the adjointness operation and the operator norm

∥a∥ = sup∥au∥ / u ∈ H, ∥u∥ ≤

this algebra is a C∗-algebra. In the nite dimensional case, one recoversMn(C). ⧫

Example .. (¿e algebra of compact operators)LetH be an Hilbert space. A nite rank operator a ∈ B is an operator such that dimRan a <∞. LetBF denote the subalgebra of nite rank operators in B. ¿e algebraK(H) = K of compact operatorsis the closure ofBF for the topology of the operator norm. ¿e algebraK is aC∗-algebra, which is notunital whenH is innite dimensional. In caseH is nite dimensionnal, K =Mn(C) for n = dimH.

For any integer n ≥ , Mn(C) is identied as a subalgebra of B, as the operators which act onlyon the rst n vectors of a xed orthonormal basis ofH. ¿en one gets a direct system of C∗-algebrasinside B, in ∶Mn(C)Mn+(C) with in(a) = ( a

). One has

K = limÐ→Mn(C)

Using this identication, it is easy to see thatK is an ideal inB. ¿e quotientC∗-algebraQ = B/Kis the Calkin algebra. ⧫

Example .. (¿e algebra of continuous functions)Let X be a compact Hausdor space. Denote by C(X) the (commutative) algebra of continuousfunctions on X, for pointwise addition andmultiplication of functions. Dene the involution f↦ fand the sup norm

∥f∥∞ = supx∈X

∣f(x)∣ (..)


With these denitions, C(X) is a C∗-algebra.If the topological space X is only a locally compact Hausdor space, one denes C(X) to be

the algebra of continous functions on X vanishing at innity : for any є > , there exists a compactK ⊂ X such that f(x) < є for any x ∈ X/K. Equipped with the same norm and involution as C(X),this is a C∗-algebra. ⧫

Example .. (¿e tensor product)One can perform a lot of operations on C∗-algebras, some of them being described in the followingexamples.

Let us mention that there exist some well dened tensor products onC∗-algebras (see Chapter in [Kadison andRingrose, ] or Appendix T in [Wegge-Olsen, ]). In the following, we denoteby ⊗ the spatial tensor product. One of its properties is that C(X) ⊗ C(Y) = C(X × Y) for anycompact spaces X,Y. ⧫

Example .. (¿e algebraMn(A))Let A be a C∗-algebra. We denote by Mn(A) the set of n × n matrices with entries in A. ¿isis naturally an algebra. One can dene the max norm and the sum norm on this algebra using∥ai j∥ instead of ∣ai j∣ as in Example ... For the operator norm, the situation is more subtle. Onehas to take an injective representation ρ ∶ A → B(H), which induces an injective representationρn ∶ Mn(A) → B(Hn). ¿e operator norm of a ∈ Mn(A) is dened as ∥a∥ = ∥ρn(a)∥ where the lastnorm is on B(Hn). One can then show that this norm is independent of the choice of the injectiverepresentation ρ and givesMn(A) a structure of C∗-algebra.

¿is construction corresponds also to deneMn(A) asMn(C) ⊗ A.¿e natural inclusionMn(A) Mn+(A) denes a direct system of C∗-algebras. One can show

that A ⊗ K = limÐ→Mn(A). ⧫

Example .. (¿e algebra C(X,A))Let X be a locally compact topological space andA a C∗-algebra. ¿e space C(X,A) of continuousfunctions a ∶ X → A vanishing at innity, equipped with the involution induced by the involutiononA and the sup norm ∥a∥∞ = supx∈X ∥a(x)∥, is a C∗-algebra. Using the spatial tensor product, onehas C(X,A) = C(X) ⊗ A.

If X is compact, we denote it by C(X,A). IfA is unital and X is compact, this algebra is unital.⧫

Example .. (¿e convolution algebra)¿e algebra L(R) for the convolution product

(f ∗ g)(x) = ∫R f(x − y)g(y)dywith the norm

∥f∥ = ∫R ∣f(x)∣dx

and equipped with the involutionf∗(x) = f(−x)

is a Banach algebra with involution but is not a C∗-algebra. ⧫


Denition .. (Fréchet algebra)A semi-norm of algebras ponA is a semi-norm on the vector spaceA for which p(ab) ≤ p(a)p(b)for any a,b ∈ A.

A Fréchet algebra is a topological algebra for the topology of a numerable set of algebra semi-norms, which is complete. ⧫Example .. (¿e Fréchet algebra C∞(M))LetM be aC∞ nite dimension locally compactmanifold. ¿en the algebraC∞(M) of dierentiablefunctions onM is an involutive algebra but is not a Banach algebra for the sup norm because it is notcomplete. Nevertheless, this algebra can be equipped with a family of semi-norms pKr,N to make ita Fréchet algebra. ¿ese semi-norms are dened as follows. For any α = (α, . . . ,αn) where αr ∈ N,let us use the compact notation Dα = ( ∂∂x)

α ⋯ ( ∂∂xn)αn with ∣α∣ = α+⋯+αn. ¿en, for any compact

subspaceK ⊂M and any integerN ≥ , dene pK,N(f) =max∣(Dα f)(x)∣ / x ∈ K, ∣α∣ ≤ N. With aincreasing numerable family of compact spacesKr ⊂M such that⋃r≥Kr =M, one gets a numerablefamily of semi-norms pKr,N for the topology of which C∞(M) is complete. ⧫Example .. (¿e irrational rotation algebra)Let θ be an irrational number. On the Hilbert space L(S), consider the two unitary operators

(Uf)(t) = eπit f(t) (V f)(t) = f(t − θ)where f ∶ S → C is considered as a periodic function in the variable t ∈ R. ¿en one has UV =eπiθVU ∈ B(L(S)). ¿e C∗-algebra Aθ generated by U and V is called the irrational rotationalgebra or the noncommutative torus.

Let us consider the Schwartz space S(Z) of sequences (am,n)m,n∈Z of rapid decay i.e. (∣m∣ +∣n∣)q∣am,n∣ is bounded for any q ∈ N. We dene the algebra A∞θ as the set of elements in Aθ whichcan be written as ∑m,n∈Z am,nUmVn for a sequence (am,n)m,n∈Z ∈ S(Z). ¿e family of semi-normspq(a) = supm,n∈Z( + ∣m∣ + ∣n∣)q∣am,n∣ gives toA∞θ a structure of Fréchet algebra.

¿is algebra admits two continuous non inner derivations δi, i = ,, dened by δ(Um) =πimUm, δ(Vn) = Vn, δ(Um) = Um and δ(Vn) = πinVn.

Using Fourier analysis, S(Z) is isomorphic to the spaceC∞(T)whereT is the two-torus. ¿ealgebraA∞θ is then the equivalent of smooth functions on the noncommutative torusAθ. ⧫

Let us mention some general results about C∗-algebras:Proposition .. (Unitarisation)Any C∗-algebra A is contained in a unital C∗-algebra A+ as a maximal ideal of codimension one.

¿econstruction of the unitarizationA+ is as follows: as a vector space,A+ = A+C; as an algebra,(a+ λ)(b+ µ) = ab+ λb+ µa+ λµ; as an involutive algebra, (a+ λ)∗ = a∗ + λ; as a normed algebra,∥(a + λ)∥ = sup∥ab + λb∥ / b ∈ A, ∥b∥ ≤ .¿eorem ..For any C∗-algebra A, there exist a Hilbert spaceH and an injective representation A → B(H). ¿enevery C∗-algebra is a subalgebra of the bounded operators on a certain Hilbert space.

¿e construction of this Hilbert space, which is not necessarily separable, is performed throughthe GNS construction. ¿is theorem implies that any C∗-algebra can be concretely realized as analgebra of operators on a Hilbert space. Obviously, the converse is not true: there are many algebrasof operators on Hilbert spaces which are not C∗-algebras.Proposition ..Any morphism between two C∗-algebras is norm decreasing.

¿e norm on a C∗-algebra is unique. An isomorphism of C∗-algebras is an isometry.


1.2.2 The Gelfand transformLet A be a unital Banach algebra. Let us use the notation z = z1 ∈ A for any z ∈ C.Denition .. (Resolvant, spectrum and spectral radius)Let a be an element in A.

¿e resolvant of a, denoted by ρ(a), is the subspace of C:

ρ(a) = z ∈ C / (a − z)− ∈ A

¿e spectrum of a, denoted by σ(a), is the complement of ρ(a) in C: σ(a) = C/ρ(a). One canshow that σ(a) is a compact subspace of C contained in the disk z ∈ C / ∣z∣ ≤ ∥a∥.

¿e spectral radius of a is dened as

r(a) = sup∣z∣ / z ∈ σ(a) ⧫

For any a ∈Mn(C), the spectrum of a contains the set of eigenvalues of a, but can contain othervalues not associated to eigenvectors.

¿e spectrum of a ∈ A depends on the algebra A. Nevertheless, we will see exceptions to that.¿e spectral radius can be computed using the relation

r(a) = limn→∞

∥an∥/n

which can look surprising at rst: on the le , the radius is dened using only the algebraic structureof A (is an element (a − z) invertible?), on the right it is related to the norm. . .

Here are some interesting results about the spectrum of particular elements.

Proposition ..If a is self-adjoint, then σ(a) ⊂ R. If a is unitary, then σ(a) ⊂ S. If a is positive, then σ(a) ⊂ R+.

One can show the following:

¿eorem .. (Gelfand-Mazur)Any unital Banach algebra in which every non zero element is invertible is isomorphic to C.

Denition .. (¿e spectrum of an algebra and the Gelfand transform)Let A be a Banach algebra. A (continuous) character on A is a non zero continuous morphism ofalgebras χ ∶ A→ C. If A is unital, we require χ(1) = .

¿e spectrum of A, denoted by ∆(A), is the set of characters of A. ¿e spectrum ∆(A) is atopological space for the topology induced by the pointwise convergence χn

n→∞ÐÐ→ χ ⇔ ∀a ∈A, χn(a)

n→∞ÐÐ→ χ(a).¿ere is a natural map A → C(∆(A)) dened by a↦ a where a(χ) = χ(a). ¿is is the Gelfand

transform of A. ⧫

So, now that we have associated to elements in a Banach algebra, and to the algebra itself, sometopological spaces, it is time to introduce some functional spaces on them! ¿e next example givesus an insight to what will happen in the general case.

Example .. (¿e spectrum of C(X))Let A = C(X) as in Example ... For any function f ∈ C(X), σ(f) is the set of values of f:σ(f) = f(X) ⊂ C. Any point x ∈ X denes a character χx ∈ ∆(A) by χx(f) = f(x), so thatX ⊂ ∆(A). ¿e topologies on X and ∆(A)makes this inclusion a continuous application. ⧫


Example .. (Unital commutative Banach algebra)What happens when the Banach algebra is unital and commutative? In that case, one can show thatthe maximal ideals in A are in a one-to-one correspondence with characters on A. Indeed, it is easyto associate to any character χ ∈ ∆(A), the maximal ideal Iχ = Ker χ. In the other direction, for anymaximal ideal I, one can show that every non zero element in the algebra A/I is invertible, whichmeans, by the Gelfand-Mazur’s theorem, that A/I = C. Associate to I the projection A → A/I. ¿isis the desired character.

¿en, one can show that ∆(A) is a compact Hausdor space. ¿e Gelfand transform connectstwo commutative unital Banach algebras A → C(∆(A)) by a continuous morphism of algebras.What is now a pleasant surprise, is that the spectrum of a in A is exactly the spectrum of a inC(∆(A)), which is the set of values of the function a on ∆(A):

σ(a) = σ(a) = a(χ) = χ(a) / χ ∈ ∆(A) ⧫

When the commutative Banach algebra is not unital, theGelfand transform realizes a continuousmorphism of algebras A → C(∆(A)). One important result is that ∆(A)+ = ∆(A+) where on thele ∆(A)+ is the one-point compactication of the topological space ∆(A) and on the right A+ isthe unitarization of A.

It is now possible to state the main theorem in this section:

¿eorem .. (Gelfand-Neumark)For any commutative C∗-algebra A, the Gelfand transform is an isomorphism of C∗-algebras.

In the unital case, one gets A ≃ C(∆(A)) and in the non unital case, A ≃ C(∆(A)) and A+ ≃C(∆(A)+).

In the language of categories, this theoremmeans that the category of locally compact Hausdorspaces is equivalent to the category of commutative C∗-algebras.

1.2.3 Functional calculus¿e demonstration of the Gelfand-Neumark theorem relies on some constructions largely knownas functional calculus. ¿ese constructions are very important to understand the relations betweencommutative C∗-algebras and topological spaces. ¿eir understanding opens the door to the com-prehension of noncommutative geometry.

¿e rst example we consider is the polynomial functional calculus. ¿is gives us the generalidea. Let a ∈ A, where A is any unital associative algebra. To every polynomial function p ∈ C[x] inthe real variable x, we can associate p(a) ∈ A as the element obtained by the replacement xn ↦ anin p. In particular, for the polynomial p(x) = x (resp. p(x) = ), one gets p(a) = a (resp. p(a) = 1).

For algebras with supplementary structures, this can be generalized using other algebras of func-tions.

First, consider an involutive unital algebra A, and let a ∈ A be a normal element. To everypolynomial function p ∈ C[z,z] of the complex variable z and its conjugate z, we associate p(a) ∈ Athrough the replacements zn ↦ an and zn ↦ (a∗)n. Because a is normal, p(a) is a well denedelement in A.

Let A be now a unital Banach algebra. For any λ ∉ σ(a) one introduces the resolvant of a at λ:

R(a, λ) = λ− a ∈ A


Consider any holomorphic function f ∶ U → C, with U an open subset of C which strictly containsthe compact subspace σ(a), and Γ ∶ [, ]→ C a closed path inU such that σ(a) is strictly inside Γ.¿e usual Cauchy formula f(z) =

πi ∫Γf(λ)λ−z dλ ∈ C can be generalized in the form

f(a) = πi ∫Γ

f(λ)λ− adλ ∈ A

Indeed, λ ↦ R(a, λ) is a function which takes its values in a Banach space, and integration off(λ)R(a, λ) is meaningful along Γ. What can be shown is that this integration does not dependon the choice of the surrounding closed path Γ.

¿is relation denes what is called the holomorphic functional calculus on A. In case f is apolynomial function (of the variable z, but not of the variable z), f(a) coincides with the polynomialfunctional calculus.

Let us consider now a unital C∗-algebra A. In that case, one would like to mix the two previousfunctional calculi on involutive and Banach algebras.

In order to do that, consider a normal element a ∈ A. One can introduce C∗(a), the smallestunital C∗-subalgebra of A which contains a and a∗ (and 1 since it is unital). Because 1,a and a∗commute among themselves, the C∗-algebra C∗(a) is a commutative C∗-algebra. Let us summarizesome facts about this algebra:

Proposition ..¿e spectrum of a in C∗(a) is the same as the spectrum of a in A. It will be denoted by σ(a).

¿e spectrum of the algebra C∗(a) is the spectrum of the element a, ∆(C∗(a)) = σ(a), so that

C∗(a) = C(σ(a))

¿eGelfand transformmaps a into the continuous function a ∶ σ(a)→ Cwhich is identity: σ(a) ∋z↦ z ∈ C.

¿e inverse of the Gelfand transform associates to any continuous function f ∶ σ(a)→ C a uniqueelement f(a) ∈ C∗(a) ⊂ A such that

∥f(a)∥ = ∥f∥∞ σ(f(a)) = f(σ(a)) ⊂ C

In particular, the norm of f(a) in C∗(a) is the norm of f(a) in A.

¿e association f ↦ f(a) in this Proposition is the continuous functional calculus associatedto the normal element a. In case f is a polynomial function in the variables z and z (resp. f isan holomorphic function), one recovers the polynomial functional calculus (resp. the holomorphicfunctional calculus).

¿ere are a lot of interesting normal elements in a C∗-algebras (self-adjoint, unitary, positive. . . )for which the continuous functional calculus is very convenient. ¿e next example illustrates such asituation.Example .. (Absolute value in A)One can associate to any element a ∈ A its absolute value using the functional calculus associated tothe normal (and positive) element a∗a. Consider the continuous function R+ ∋ x ↦ f(x) = ∣x∣/and dene ∣a∣ ∈ A+ by ∣a∣ = f(a∗a). ⧫

What do we learn from these constructions? ¿e main result here is that it is not necessary toconsider a commutative C∗-algebra in order to manipulate some topological spaces. Just consider


somenormal elements commuting among themselves, build upon them the smallestC∗-algebra theygenerate, and you have in hand a Hausdor space!

¿e idea of noncommutative topology is to study C∗-algebras from the point of view that theyare “continuous functions on noncommutative spaces”. In order to do that, one needs some toolsthat are common to the topological situation and to the algebraic one.

Such tools exist! One of them is K-theory.

1.3 K-theory for beginners

¿e K-theory groups are dened through a universal construction, the Grothendieck group associ-ated to an abelian semigroup.

Denition .. (Grothendieck group of an abelian semigroup)An abelian semigroup is a set V equipped with an internal associative and abelian law ⊞ ∶ V×V→ V.A unit element is an element such that v ⊞ = v for any v ∈ V. Any abelian group is a semigroup.

¿e Grothendieck group associated to V is the abelian group (Gr(V),+) which satises the fol-lowing universal property. ¿ere exits a semigroup map i ∶ V → Gr(V) such that for any abeliangroup (G,+) and any morphism of abelian semigroups ϕ ∶ V → G, there exists a unique morphismof abelian groups ϕ ∶ Gr(V)→ G such that ϕ = ϕ i. ⧫

¿is means that the following diagram can be completed with ϕ to get a commutative diagram:

Gr(V)ϕ

""V

i

OO

ϕ // G

It is convenient to have in mind one of the possible constructions of the Grothendieck groupassociated to V. On the set V × V consider the equivalence relation: (v,v) ∼ (v′,v′) if there existsv ∈ V such that v ⊞ v′ ⊞ v = v′ ⊞ v ⊞ v ∈ V. Denote by ⟨v,v⟩ an equivalence class in V × V for thisrelation and let Gr(V) = (V × V)/ ∼. ¿e group structure on Gr(V) is dened by ⟨v,v⟩ + ⟨v′,v′⟩ =⟨v ⊞ v′,v ⊞ v′⟩, the unit is ⟨v,v⟩ for any v ∈ V, the inverse of ⟨v,v⟩ is ⟨v,v⟩. ¿e morphism ofabelian semigroups i ∶ V → Gr(V) is v ↦ ⟨v ⊞ v′,v′⟩ (independent of the choice of v′). Notice that⟨v + v,v + v⟩ = ⟨v,v⟩ in Gr(V).

¿en one can show that Gr(V) is indeed the Grothendieck group associated to V and that

Gr(V) = i(v) − i(v′) / v,v′ ∈ V

A useful relation is that i(v +w) − i(v′ +w) = i(v) − i(v′) ∈ Gr(V) for any w ∈ V.

Example .. (¿e semigroup N)¿e set of natural numbers denes an abelian semigroup (N,+). Its Grothendieck’s group is (Z,+).In this situation, the morphism i ∶ N→ Z is injective. ¿is is not always the case. Another particularproperty is that any relation n+n = n+n inN can be simplied into n = n. ¿is is the simplicationproperty, which is not satised by all abelian semigroups. ⧫

Example .. (¿e semigroup N ∪ ∞)Consider the set N ∪ ∞ with the ordinary additive law for two elements in N and the new law∞+ n =∞+∞ =∞. ¿en its Grothendieck group is 0. Indeed, all couples (n,m), (n,∞), (∞,m)and (∞,∞) are equivalent. ⧫


1.3.1 The topological K-theoryIt is useful to recall some facts and constructions about vector bundles over topological spaces. Wewill restrict ourselves to locally trivial complex vector bundles over Hausdor spaces.

Denition ..Let π ∶ E → X and π′ ∶ E′ → X two vector bundles over X, of rank n and n′. ¿en one denes theWhitney sum E ⊕ E′ → X of rank n + n′ by E ⊕ E′ = ∪x∈X(Ex ⊕ E′x) = (e, e′) ∈ E × E′ / π(e) =π′(e′) ⊂ E × E′ and the tensor product E⊗ E′ = ∪x∈X(Ex ⊗ E′x) of rank nn′.

We denote by Cn = X ×Cn → X the trivial vector bundles of rank n.For any continuous map f ∶ Y → X and any vector bundle E → X, we dene the pull-back

f∗E → Y as f∗E = (y, e) ∈ Y × E / f(y) = π(e) ⊂ Y × E. ⧫

When i ∶ Y X is an inclusion, the pull-back i∗E = E∣Y is just the restriction of E to Y ⊂ X.When f, f ∶ Y → X are homotopic, the two vector bundles f∗ E and f∗ E are isomorphic.

We will use the following very important result in the theory of vector bundles:

¿eorem .. (Serre-Swan)Let X be a compact topological space. For any vector bundle E → X there exist an integer N and asecond vector bundle E′ → X such that E⊕ E′ ≃ CN.

We introduce V(X), the set of isomorphic classes of vector bundles over X. Let use the notation[E] for the isomorphic class of E. ¿e set V(X) is an abelian semigroup for the law induced by theWhitney sum: [E] + [E′] = [E⊕ E′].

For any continuous map f ∶ Y → X, the pull-back construction denes a morphism of abeliansemigroups f∗ ∶ V(X)→ V(Y), which depends only on the homotopic class of f.Denition .. (K(X) for X compact)For any compact topological space X, we dene K(X) as the Grothendieck group of V(X). ⧫

Remark .. (Representatives in K(X))From the construction of the Grothendieck group, any element in K(X) can be realized as a formaldierence [E] − [F] of two isomorphic classes in V(X). Adding the same vector bundle to E and Fdoes not change this element in the Grothendieck group. So, one can always nd a representative ofthe form [E] − [Cn] for some integer n. ⧫

For any continuous map f ∶ Y → X, the morphism f∗ ∶ V(X) → V(Y) induces a morphism ofabelian groups f♯ ∶ K(X)→ K(Y).Example .. (K(∗) = Z)Let ∗ denote the space reduced to a point. In that case any vector bundle E → ∗ is just a nitedimensional vector space. It is well known that the isomorphic classes of nite dimensional vectorspaces are classied by their dimension, so that V(∗) = N, with the abelian semigroup structure ofExample ... ¿en one getsK(∗) = Z. Obviously this result is true for any contractible topologicalspace. ⧫

Let x ∈ X be a xed point. Denote by i ∶ ∗ = x → X the inclusion, and p ∶ X → ∗ theprojection. ¿en p i is Id∗. ¿ese maps dene two morphisms

p♯ ∶ Z = K(∗)→ K(X) i♯ ∶ K(X)→ K(∗) = Z

Because i♯p♯ = IdZ ∶ Z→ Z, p♯ is injective.

1.3 K-theory for beginners 25

Denition .. (Reduced K-group for pointed compact spaces)We dene the reduced K-group of the pointed compact topological space X by

K(X) = Ker(i♯ ∶ K(X)→ Z) = K(X)/p♯Z ⧫

¿e injective morphism p♯ splits the short exact sequence of abelian groups

0 //K(X) //K(X) i♯ //K(∗) = Z //0

so that K(X) = K(X) ⊕ Z. ¿e reduced K-theory, like the reduced singular homology, is thenatural K-theory of pointed compact spaces.

Remark .. (Interpretation of K(X))An element in K(X) is a formal dierence [E]−[F]where now E and F have the same rank becausethey must coincide over x in order to be in the kernel of i♯. It is possible to choose F = CN withN = rank E. ⧫

Denition .. (K(X) for any X)Let X be a locally compact topological space X, not necessarily compact. Denote by X+ its one-pointcompactication. ¿en one denes K(X) = K(X+). ⧫

Let us make some comments about this construction.Remark ..In this situation, the natural xed point in X+ is the point at innity, so that i ∶ ∗ → X+ sends ∗into∞. ¿e compactication adds a point to X and the reduced K-group construction removes thecontribution from this point.

In case X is compact, it is then easy to verify that K(X+) identies with the K-group as inDenition ... ⧫

Remark .. (Interpretation of K(X))In K(X+), an element is a formal dierence [E] − [F] with rank E = rank F. Because this elementis in the kernel of i♯, one has E∣∞ = F∣∞. So these two vector bundles are also isomorphic in aneighborhood of ∞ ∈ X+. By denition of the one-point compactication, such a neighborhoodis the complement of a compact in X, so that E and F can be considered as vector bundles over Xwhich coincide outside some compact K ⊂ X.

One can go a step further. Because E and F coincide outside some compact K, one can add tothem a third vector bundle such that outside K the sums are isomorphic to a trivial vector bundle.Adding such a vector bundle does not change the element [E]−[F] ∈ K(X). ¿erefore, any elementin K(X) can be represented by a formal dierence [E]−[F]where E and F are not only isomorphicoutside some compact, but also trivial.

Owing to this interpretation, this denition of K(X) = K(X+) is also called, in the literature,the K-theory with compact support (the non trivial part of the vector bundles is inside a compact).For instance, it is denoted by Kcpt in [Lawson and Michelsohn, ]. ⧫

¿e indice in the denition of the K-group suggests that others K-groups can be dened. ¿isis indeed the case, but we will see that there are not so many!

Denition .. (Higher orders K-groups)Let X be a locally compact topological space X. For any n ≥ , we dene K−n(X) = K(X ×Rn). ⧫


¿e corresponding reduced K-group is K−n(X) = K(X ∧ Sn), where we recall that for twopointed compact spaces (X,x) and (Y, y), their wedge product is X∧Y = X×Y/(x×Y ∪X×y). For any n, one can show that K−n(X) = K−n(X+).Proposition .. (Long exact sequences)Let X be a locally compact space and Y ⊂ X a closed subspace. ¿en there exist boundary mapsδ ∶ K−n(Y)→ K−n+(X/Y) and a long exact sequence

⋯ δ //K−n(X/Y) //K−n(X) //K−n(Y) δ //K−n+(X/Y) //⋯

⋯ δ //K(X/Y) //K(X) //K(Y)

In reduced K-theory, for pointed compact spaces Y ⊂ X, one has the corresponding long exactsequence

⋯ δ //K−n(X/Y) //K−n(X) //K−n(Y) δ //K−n+(X/Y) //⋯

⋯ δ //K(X/Y) //K(X) //K(Y)

Proposition .. (¿e ring structure)¿e tensor product of vector bundles induces a ring structure on K(X) and K(X).

¿e external tensor product induces graded ring structures on K(X) = ⊕n≥K−n(X) and onK(X) =⊕n≥ K−n(X) which extend the ring structures on K(X) and K(X).

Example .. (¿e -sphere)Any vector bundle on the -sphere is characterized by its clutching function on the equator. ¿isis a continuous map S → U(n) for a vector bundle of rank n. In order to consider all the possibleranks at the same time, the maps to consider are S → U(∞) = limÐ→U(n). Studying these func-tions, in particular their homotopic equivalence classes, gives the following result. LetH denote thetautological vector bundle of rank over CP = S. ¿en one has (H ⊗ H) ⊕ C ≃ H ⊕ H and asrings K(S) ≃ Z[H]/⟨(H − C)⟩ where ⟨(H − C)⟩ is the ideal in Z[H] generated by (H − C),so that K(S) ≃ Z ⋅ C ⊕ Z ⋅ (H − C) ≃ Z ⊕ Z. Because H − C ∈ Ker(i♯ ∶ K(S) → Z), one hasK(S) = Z ⋅ (H −C) ≃ Z with a null product. ⧫

Example .. (¿e ring K(∗))¿e ring structure of K(∗) is easy to describe. One can show that K−(∗) = K(R) = K(S) = Z.Denote by ξ the generator of K−(∗). ¿en, one can show that K(∗) = Z[ξ]. As ξ is of degree −,one has K−n(∗) = Z and K−(n+)(∗) = 0. ⧫

Here is now the main result in K-theory:

¿eorem .. (Bott periodicity)For any locally compact space X, one has a natural isomorphism

K(X ×R) = K−(X) ≃ K(X)

For any pointed compact space X, one has a natural isomorphism

K(X ∧ S) = K−(X) ≃ K(X)


In reduced K-theory, for two pointed compact spaces X,Y, there is a natural (graded) product

K(X)⊗ K(Y)→ K(X ∧ Y)

Using Y = S, this product gives us an isomorphism

K(X)⊗ K(S) ≃→ K(X ∧ S)

which is exactly the Bott periodicity. Indeed, we saw in Example .. that K(S) is generated byH −C (with (H −C) = ). ¿e Bott periodicity is the isomorphism

β ∶ K(X) ≃→ K(X ∧ S) = K−(X)a↦ (H −C) ⋅ a

Example .. (K-theories of spheres)One has Sn ∧ Sm ≃ Sn+m, so that K(Sn) = K(Sn− ∧ S) = K(S) = Z. For odd degrees, one onlyneeds to know K(S). Using standard arguments from topology of ber bundles (see [Steenrod,] for instance), there are no non trivial (complex) vector bundles over S, so that V(S) = N andthen K(S) = Z and K(S) = 0. ¿is shows that K(Sn+) = K(S) = 0. Notice that becauseR+ = S (one-point compactication), one has K(R) = 0. ⧫

Proposition .. (Six term exact sequences in K-theory)¿e Bott periodicity reduces the long exact sequences of Proposition .. into two six term exact se-quences

K(X/Y) // K(X) // K(Y)δ

K−(Y)δ

OO

K−(X)oo K−(X/Y)oo

(..)

for locally compact spaces Y ⊂ X with Y closed, and

K(X/Y) // K(X) // K(Y)δ

K−(Y)

δ

OO

K−(X)oo K−(X/Y)oo

for pointed compact spaces Y ⊂ X with Y closed.

Example .. (K-groups for some topological spaces)Here is a table of some known K-groups for ordinary topological spaces.

Topological space K K−

∗, compact contractible Hausdor space Z 0], ] 0 0

R, ], [ 0 ZRn, n ≥ Z 0

Rn+, n ≥ 0 ZSn, n ≥ Z⊕Z 0

Sn+, n ≥ Z ZTn Zn− Zn−


Remark .. (Real topological K-theory)We have introduced the topological K-theory using the complex vector bundles over topologicalspaces. It is possible to dene a real topological K-theory in exactly the same way using real vectorbundles. ¿e theory is dierent. For instance, there are non trivial real vector bundles over S (thinkat the Moebius trip), but there are no non trivial complex vector bundles. In real K-theory the Bottperiodicity is of period , and the six term exact sequence is replaced by a terms exact sequence.⧫

Remark .. (¿e origin of Bott periodicity)¿e rst paper mentioning Bott periodicity, [Bott, ], was concerned with the homotopy of clas-sical groups, in particular U(n) and O(n). What Bott discovered is that if one denotes by U(∞) =limÐ→U(n) and O(∞) = limÐ→O(n) the inductive limits for the natural inclusions U(n) U(n + )and O(n) O(n + ), then

πk(U(∞)) = πk+(U(∞)) πk(O(∞)) = πk+(O(∞))

In fact, for n large enough, πk(U(n)) = πk(U(n + )) for n > k/, so that this periodicity expressesitself before innity. ¿e period for the complex case U(∞) (resp. for the real case O(∞)) isrelated to the period for K-theory (resp. real K-theory). See [Karoubi, ] for a review andreferences. ⧫

1.3.2 K-theory for C∗-algebrasTopological K-theory is dened using some explicit geometrical constructions on vector bundlesover a compact topological space X. ¿ese constructions, except the tensor product of vector bun-dles, can be described using the C∗-algebra C(X).

Indeed, it is a well known fact that continuous sections of a vector bundle E → X is a C(X)-module. Recall the following denitions about modules. From now on, every modules are le mod-ules.Denition .. (Finite projective modules)Let A be a unital associative algebra.

M is a free A-module if it admits a free basis.M is a projective A-module if there exists a A-module N such thatM⊕ N is a free module.M is a nite projectiveA-module if there exist aA-moduleN and an integerN such thatM⊕N ≃

AN . ⧫

¿eorem .. can then be written in the following algebraic form:

¿eorem .. (Serre-Swan, algebraic version)¿e functor “continuous sections” realizes an equivalence of categories between the category of vectorbundles over a compact topological space X and the category of nite projective modules over C(X).

Anynite projectivemodule is characterized by themorphismofA-modules p ∶ AN → AN whichprojects onto M. ¿is morphism is representable as a projection p ∈ MN(A), p = p, p∗ = p, suchthat M = ANp. In particular, any vector bundle over X is given by a projection p ∈ MN(C(X)) =C(X,Mn(C)) (see Example ..).

We have dened the topological K-theory via the isomorphic classes of vector bundles. In thealgebraic language, isomorphic classes correspond to some equivalence classes on projections. Letus dene some possible equivalence relations on projections in C∗-algebras.

Let us denote by P(A) = p ∈ A / p = p∗ = p the set of projections in a unital C∗-algebra A.


Denition .. (Equivalences of projections)A partial isometry is an element v ∈ A such that v∗v ∈ P(A). In that case, one can show that vv∗ ∈P(A). An isometry is an element v ∈ A such that v∗v = 1. Unitaries are in particular isometries.

Two projections p,q ∈ P(A) are orthogonal if pq = qp= ∈ A. ¿is means that they project ondirect summands of A. In this situation p⊕ q ∈ P(A) is well dened.

¿ere are three notions of equivalence for two projections p,q ∈ P(A):

homotopic equivalence: p∼h q if there exists a continuous path of projections inA connectingpand q.

unitary equivalence: p∼u q if there exists a unitary element u ∈ A such that u∗pu = q.

Murray-von Neumann equivalence: p∼M. v.N. q if there exists a partial isometry v ∈ A such thatv∗v = pand vv∗ = q. ⧫

One can show thatp∼h qÔ⇒ p∼u qÔ⇒ p∼M. v.N. q

Dene Pn(A) ⊂ Mn(A), the set of projections in Mn(A). ¿e natural inclusions in ∶ Mn(A) Mn+(A)with in(a) = ( a

) permits one to deneM∞(A) = ⋃n≥Mn(A) andP∞(A) = ⋃n≥Pn(A).¿e three equivalence relations dened above are well dened on P∞(A).Proposition .. (Stabilisation of the equivalence relations)In P∞(A), the three equivalence relations coincide.

We will denote this relation by ∼.Denition .. (K(A) for unital C∗-algebra)Let V(A) denote the set of equivalence classes in P∞(A) for the relation ∼. ¿is is an abelian semi-group for the addition p⊕ q = ( p

q) ∈ P∞(A).¿e group K(A) is the Grothendieck group associated to (V(A),⊕). ⧫

Example .. (K(Mn(C)))Let us look at the algebra A = C. In that case, a projection p ∈ P∞(C) is represented by a projectionp ∈ MN(C) for a suciently large N. Such a projection denes the vector space Ran p ⊂ CN ofdimension rank p. It is easy to see that the equivalence relation ∼ detects only this dimension, sothat V(C) = N and K(C) = Z.

Let us consider now A = Mn(C). One has MN(Mn(C)) = MNn(C), so that P∞(Mn(C)) =P∞(C), with the same equivalence relation. ¿en one hasK(Mn(C)) = Z. ¿is result is an exampleof Morita invariance of the K-theory. ⧫

More generally, by the same argument, one can show:

Proposition .. (Morita invariance of the K-theory)

K(Mn(A)) = K(A)

Example .. (K(C(X)))Let X be a compact topological space. ¿en by¿eorem .. one has

K(C(X)) = K(X) ⧫


Any morphism of C∗-algebras ϕ ∶ A → B gives rise to a natural map ϕ ∶ P∞(A) → P∞(B)compatible with the relation ∼ on both sides. ¿is induces a morphism of semigroups V(A)→ V(B)and a morphism of abelian groups ϕ♯ ∶ K(A)→ K(B).

When the algebra A is not unital, consider its unitarization A+. ¿en one has the short exactsequence of C∗-algebras

0 //A i //A+π //C //0

Denition .. (K(A) for non unital C∗-algebra)For a non unital algebra A, one denes

K(A) = Ker(π♯ ∶ K(A+)→ K(C) = Z) ⧫

Remark ..Exactly as in Remark .., this construction adds a point (the unity) and removes its contributiona erwards. In case A is unital, one can show that the two denitions coincide. More generally, asabelian groups, one has K(A+) = K(A)⊕Z. ⧫

Remark .. (Interpretation of K(A))An element in K(A) is a dierence [p] − [q] for some projections p,q ∈ Pn(A+), for large enoughn, such that [π(p)] − [π(q)] = . In fact, it is possible to choose pand q such that p− q ∈Mn(A) ⊂Mn(A+) (p− q is not a projection in this relation!). Adding a common projection, one can alwaysrepresent an element in K(A) as [p] − [1n], where 1n ∈Mn(A+) is the unit matrix. ⧫

Example .. (K(B))For any integer n and innite dimensional separable Hilbert spaceH, one hasMn(B) ≃ B (becauseHn ≃ H here), so we only need to consider projections in B. Two projections in B are equivalentprecisely when their ranges are isomorphic. So that only the dimension (possibly innite) is aninvariant, and one gets V(B) = N ∪ ∞, which produces K(B) = 0 (see Example ..). ⧫

Denition .. (Higher orders K-groups)Let A be a C∗-algebra. ¿e suspension of A is the C∗-algebra SA = C(R,A) (see Example ..).

For any n ≥ , we dene Kn(A) = K(SnA) where SnA = S(Sn−A) is the n-th suspension ofA.⧫

¿is denition leads to the following useful result:

Proposition .. (Long exact sequences)For any short exact sequence of C∗-algebras

0 //I //A //A/I //0

there exist boundary maps δ ∶ Kn(A/I)→ Kn−(I) and a long exact sequence

⋯ δ //Kn(I) //Kn(A) //Kn(A/I) δ //Kn−(I) //⋯

⋯ δ //K(I) //K(A) //K(A/I)

Remark .. (Other denition of K(A))Let us introduce the following groups


• GLn(A+), invertible elements inMn(A+)

• GL+n(A) = a ∈ GLn(A+) / π(a) = 1n where π ∶ A+ → C is the projection associated to theunitarization

• Un(A+), unitaries inMn(A+)

• U+n (A) = u ∈ Un(A+) / π(u) = 1n

¿ese groups dene some direct systems for the natural inclusion g↦ ( g 1 ). Denote by GL∞(A+),

GL+∞(A), U∞(A+) and U+∞(A) their respective inductive limits.One can show that for any C∗-algebra A, one has

K(A) = GL∞(A+)/GL∞(A+) = GL+∞(A)/GL+∞(A)= U∞(A+)/U∞(A+) = U+∞(A)/U+∞(A)

where the index means the connected component of the unit element. ⧫

Proposition .. (Continuity for direct systems)Let (Ai,αi) be a direct system of C∗-algebras. ¿en for any n one has Kn(limÐ→Ai) = limÐ→Kn(Ai).

Example .. (K(K))¿e algebra of compact operators is the direct limitK = limÐ→Mn(C). AsK(Mn(C)) = Z is a station-ary system, one has K(K) = Z. Explicitly, the isomorphism is realized as the trace [p]↦ Tr(p). ⧫

More generally we have:

Proposition .. (Morita invariance of K-theory)For any C∗-algebra A, and any n, one has Kn(A ⊗ K) = Kn(A).

Here is the version of Bott periodicity for K-theory of C∗-algebras:

¿eorem .. (Bott periodicity)For any C∗-algebra A, one has

K(SA) = K(A) ≃ K(A)

Proposition .. (Six term exact sequence)¿e Bott periodicity theorem reduces the long exact sequence associated to any short exact sequence ofC∗-algebras to a six term exact sequence

K(I) // K(A) // K(A/I)δ

K(A/I)δ

OO

K(A)oo K(I)oo

Remark .. (K-groups via homotopy groups)We have seen in Remark .. that the K-group can be dened using the -th homotopy group asK(A) = π(U∞(A+)). It is possible to show that more generally

Kn(A) = πn−(U∞(A+))


Bott periodicity is then directly equivalent to

πn+(U∞(A+)) ≃ πn(U∞(A+))

Because Un(A+) and GLn(A+) have the same topology (one is the retraction of the other), theserelations make sense with GL∞(A+). ⧫

Example .. (K-groups for some C∗-algebras)Here is a table of some known K-groups for ordinary C∗-algebras.

Algebra K K

C,Mn(C), K(H) (compacts op.) Z 0B(H) (bounded op.) 0 0Q(H) (Calkin’s alg.) 0 ZT (Tœplitz’ alg.) Z 0

On, n ≥ (Cuntz’ alg.) Zn− 0Aθ, θ irrationnal Z ≃ θZ +Z Z

C∗(Fn) (Fn free group with n generators) Z Zn

Mn(A), A ⊗ K (stabilisation) K(A) K(A)A+ (unitarization) K(A)⊕Z K(A)

SA = C(], [,A) (suspension) K(A) K(A)CA = C(], ],A) (cone) 0 0

Remark .. (K-theory computed on dense subalgebras)For a lot of examples, one can compute the K-groups of a C∗-algebra A using a dense subalgebra B.For instance, for any compact nite dimensional manifold M, the K-theory of C(M) (continuousfunctions) is the same as the K-theory of the Fréchet algebra C∞(M). ¿e same situation occurs forthe irrational rotation algebra: Kn(Aθ) = Kn(A∞θ ).

In the geometric situation, it is possible to understand this result. Smooth structures are su-ciently dense in continuous structures: any continuous vector bundle can be deformed into a smoothone. . .

Here is a description of somemore general situations. LetA be C∗-algebra (or a Banach algebra)and A∞ ⊂ A a dense subalgebra (but not necessarily a C∗-subalgebra). ¿e exponent∞ does notmean that we consider “dierentiable” functions, even if in practice this can happen: think aboutA∞θ ⊂ Aθ as a typical example. Let A+ and A∞

+ their unitarizations. Suppose that A∞+ is stable under

holomorphic functional calculus, which means that for any a ∈ A∞+ and any holomorphic function

f in a neighborhood of the spectrum of a, f(a) ∈ A∞+ .

Using the topologies induced on A∞+ and GLn(A∞

+ ) by the topologies on A+ and GLn(A+), itis possible to dene K-groups by using the relations in Remark ... ¿en one has the densitytheorem: the inclusion i ∶ A∞ → A induces isomorphisms

i♯ ∶ Kn(A∞) ≃→ Kn(A)

for any n ≥ . ⧫

Remark .. (K-homology)As for many other ordinary homologies, there exists a dual version of the K-theory of C∗-algebras,named K-homology, which we outline here.


A Fredholm module over the C∗-algebra A is a triplet (H,ρ,F) where H is a Hilbert space, ρis an involutive representation of A in B(H), and F is an operator on H such that for any a ∈ A,(F − )ρ(a), (F − F∗)ρ(a) and [F,ρ(a)] are in K. Such a Fredholm module is called odd.

AZ-graded Fredholmmodule is a Fredholmmodule (H,ρ,F) such thatH = H+⊕H− and ρ(a)is of even parity in this decomposition, and F is of odd parity: ρ(a) = ( ρ+(a)

ρ−(a) ) and F = ( U+U− ).

With these notations, U± ∶ H∓ → H± are essentially adjoint (adjoint modulo compact operators).Such a Fredholm module is called even.

In the even case, one has a natural grading map γ ∶ H → H dened by γ = ( − ) on the decom-

positionH = H+ ⊕H−. It satises γ = γ∗, γ = , γρ(a) = ρ(a)γ and γF = −Fγ.Two Fredholmmodules (H,ρ,F) and (H′,ρ′,F′) are unitary equivalent if there exists a unitary

map U ∶ H′ → H such that ρ′ = U∗ρU and F′ = U∗FU. ¿is denes an equivalence relation ∼U ofFredholm modules.

A homotopy of Fredholm modules is a familly t↦ (H,ρ,Ft) with [, ] ∋ t↦ Ft continuous forthe operator norm inH. Two Fredholmmodules are homotopic equivalent if they are connected bya homotopy of Fredholm modules. ¿is denes an equivalence relation ∼h.

¿e direct sum of two Fredholm modules (H,ρ,F) and (H′,ρ′,F′) is dened by

(H,ρ,F)⊕ (H′,ρ′,F′) = (H ⊕H′,( ρ ρ′ ) ,( F

F′ ))

¿eK-homology groupK(A) ofA is theGrothendieck group of the abelian semigroup of equiv-alence classes of even Fredholm modules for ∼U and ∼h. ¿e unit for the addition is the class of theFredholm module (,,), the inverse of the class of (H,ρ,F) is the class of (H,ρ,−F).

¿e K-homology group K(A) is dened in the same manner using odd Fredholm modules.A degenerated Fredholm module is a Fredholm module for which (F − F∗)ρ(a) = , (F −

)ρ(a) = and [F,ρ(a)] = for any a ∈ A. ¿e equivalence class of such a Fredholm module iszero.

In each equivalence class, there is a representative for which F∗ = F (self-adjoint Fredholmmod-ule) and F = (involutive Fredholm module).

For A = C, the representation ρ denes a projection p = ρ() on H, and one can show thatmodulo compact operators, one has (H,ρ,F) = (pH,ρ, pFp)⊕ (( − p)H,ρ,( − p)F( − p)). ¿erepresentation for the second Fredholm module is zero, so that its class is zero. pFp is an ordinaryFredholm operator onH, and its index induces an isomorphism Ind ∶ K(C) ≃Ð→ Z.

For any C∗-algebra A, let pbe a projection in Pn(A), and (H,ρ,F) a Fredholm module. ¿en,in the Hilbert space ρ(p)(H⊗Cn), the operator ρ(p)(F ⊗ 1n)ρ(p) is a Fredholm operator, and itsindex denes a pairing between K(A) and K(A): ⟨[p], [(H,ρ,F)]⟩ = Ind ρ(p)(F ⊗ 1n)ρ(p) ∈ Z.

For more developments in K-homology, see [Blackadar, ] and [Higson and Roe, ]. ⧫

1.3.3 Algebraic K-theoryUntil now, K-theory has been dened using topological structures, either at the level of a space or atthe level of an algebra (remember that they are the manifestations of the same topological structurein the commutative case).

Nevertheless the group K(A) can be dened in the pure algebraic context. Indeed, to any ringA, which we take unital from now on, one can associate its category of nite projective modules.

Denition .. (Kalg (A) for unital ring A)

¿e group Kalg (A) is the Grothendieck group associated to the semigroup of isomorphic classes of

nite projective modules onA, on which the additive law is induced by the direct sum of modules.⧫


As in the topological case, every nite projectiveA-moduleM is characterized by a (non unique)projector p ∈Mm(A). Be aware of the dierent terminologies that are used. “Projection” is reservedto the C∗-algebra context, because in that case p satises p = p and p∗ = p. “Projector” is moregeneral, in that case p satises only p = p.

¿e equivalence relation we use on these projectors is the following:

Denition .. (Equivalence relation on projectors)Two projectors p ∈ Mm(A) and q ∈ Mn(A) are equivalent if there exist an integer r ≥ m,n and aninvertible u ∈ GLr(A) such that p,q ∈ Mr(A) are conjugated by u: p = u−qu. We donote by ∼ thisequivalence relation. ⧫

If p∼ q, then they dene isomorphic nite projective modules.In case A is a C∗-algebra, one can show that in the equivalence class of any projector pone can

nd a projection. ¿is means that the two semigroups which dene the K-theories are the same :

Kalg (A) = K(A) (as a C∗-algebra)

For higher order groups, the situation is no more equivalent. ¿e Denition .. (or theirequivalent ones given in Remark ..) uses extensively the topological structure of the algebra,either to dene continuous functions R → A or to compute the homotopy groups of the spacesU∞(A+) in Remark ...

Nevertheless, one can dene Kalg (A) as follows:

Denition .. (Kalg (A) for unital ring A)

One denes

Kalg (A) = GL∞(A)/[GL∞(A),GL∞(A)] = GL∞(A)ab ⧫

Let A be a C∗-algebra. A well known fact about invertibles is that if u,v ∈ GL∞(A) then uv andvu are homotopic. So that there is a natural morphism of groups

Kalg (A)→ K(A) (as a C∗-algebra)

which factors out by the homotopic relation.For every n ≥ , there is a denition which surprisingly uses some topological objects: Kalg

n (A)is the πn group of a topological space associated to the classifying space BGL∞(A) where GL∞(A)is considered as a discrete group.

In algebraic K-theory, there is no Bott periodicity, but there are some other beautiful and pow-erful results which are beyond the scope of this introduction: in the following, we will only makeuse of Kalg

(A) and Kalg (A). For a review, see [Karoubi, ] or [Rosenberg, ].

1.4 Cyclic homology for (differential) geometers

Now we have in hand some tools to characterize noncommutative topological spaces. But topologyis not everything in life. Dierential geometry has to be considered also! In this section, we willexplore other concepts that look very much like dierential forms.

1.4 Cyclic homology for (differential) geometers 35

1.4.1 Differential calculiDierential forms on a dierentiablemanifold dene a dierential graded commutative algebra. ¿isconcept can be generalised:

Denition .. (Dierential calculus on an algebra)Let A be an associative algebra. A dierential calculus on A is a graded dierential algebra (Ω,d)such that Ω = A. ⧫

Remember the denition of a graded dierential algebra: Ω is a graded algebra on which thedierential satises d(ωη) = (dω)η+ (−)∣ω∣ω(dη) for any ω,η ∈ Ω and where ∣ω∣ is the degree ofω.

In this denition, one does not suppose this graded algebra to be a graded commutative algebra.

Example .. (de Rham dierential calculus)Let M be a nite dimensional dierential manifold. ¿e graded dierential algebra (Ω(M),d) ofdierential forms is a dierential calculus on C∞(M). ⧫

¿ere are many possibilities to dene a dierential calculus on an algebra. One can summarizethe questing a er noncommutative dierential geometry to the search of some reasonable denitionof such a dierential calculus on any algebra. Many propositions have been made, depending onthe context: associative algebra without any additional structure, topological or involutive algebras,quantum groups. . .

Some examples will be given a er we introduce three main examples which are the universaldierential calculi.

Example .. (Universal dierential calculus for associative algebra)¿is dierential calculus is dened to be the free graded dierential algebra generated by A as ele-ments in degree . It is denoted by (Ω(A),d).

Because it is freely generated, it has the following universal property: for any dierential calculus(Ω,d) on A, there exists a unique morphism of dierential calculi ϕ ∶ Ω(A) → Ω (of degree )such that ϕ(a) = a for any a ∈ A = Ω(A) = Ω.

¿is implies that if (Ω,d) is generated (possibly with relations) by A = Ω then it is a quotientof (Ω(A),d) by a dierential two-side ideal.

Concretely, any element inΩn(A) is a sum of terms either of the form adb . . .dbn or of the formdb . . .dbn. ¿is property gives us an identication of le A-modules

Ωn(A) = A+ ⊗A⊗n

by the morphism adb . . .dbn ↦ ( + a) ⊗ b ⊗ ⋅ ⋅ ⋅ ⊗ bn and db . . .dbn ↦ ( + ) ⊗ b ⊗ ⋅ ⋅ ⋅ ⊗ bn,where ( + a) and ( + ) are elements in A+ = C⊕A. Be aware of the fact that this identication isnot an identication of graded dierential algebras, neither of bimodules. ⧫

In the dierential calculus (Ω(A),d), if A is unital, d1 is not zero, because it is identied with⊗ 1 ∈ A+ ⊗A. It is the aim of the following example to show that in the more restrictive situationwhere A is unital, one can promote the unit of A to a unit of the dierential calculus.

Example .. (Universal dierential calculus for associative unital algebra)LetA be an associative unital algebra. ¿e dierential calculus (Ω

U(A),dU) is dened to be the freeunital graded dierential algebra generated by A in degree . ¿e indice U stands for unital.


Because this algebra is required to have an unit, this unit is necessarily the unit in A = ΩU(A).

¿en the derivative law for dU gives dU1 = . ¿is dierential calculus admits a universal propertyas the previous one does: for any unital dierential calculus (Ω,d) on A, there exists a uniquemorphism of unital dierential calculi ϕ ∶ Ω

U(A) → Ω (of degree ) such that ϕ(a) = a for anya ∈ A = Ω

U(A) = Ω.Because (Ω

U(A),dU) is a dierential calculus generated by A, it is a quotient of (Ω(A),d).¿is quotient reveals itself in the concrete identication of Ω(A): any element in Ωn

U(A) is a sumof terms of the form adUb⋯dUbn. Here a can be 1, and in this case 1dUb⋯dUbn = dUb⋯dUbn. Ifone of the bk is proportional to 1, one has adUb⋯dUbn = . ¿is leads to the identication of le modules

ΩnU(A) = A⊗A

⊗n

by the map adUb . . .dUbn ↦ a ⊗ b ⊗ ⋅ ⋅ ⋅ ⊗ bn, where b is the projection of b ∈ A onto the vectorspace A = A/C1. ⧫

In these two examples, even if the algebra A is commutative, the graded algebras are not gradedcommutative. For commutative algebras, it is possible to construct a dierential calculus with agraded commutative algebra.

Example .. (Kähler dierential calculus for commutative unital algebra)Let A be an associative commutative unital algebra over a eld K. ¿e Kähler dierential calculus(Ω

A∣K,dK) is dened to be the free unital graded commutative dierential algebra generated by Ain degree .

One can show that the algebra ΩA∣K is an exterior algebra over A: Ω

A∣K = ⋀AΩ

A∣K. Moreover,let I ⊂ A ⊗ A be the kernel of the product map µ ∶ A ⊗ A → A. Consider A ⊗ A as an algebra(commutative) and introduce I, generated by the products of elements in I. ¿en one has the explicitconstruction Ω

A∣K = I/I.We denote by π ∶ Ω(A)→ Ω

A∣C the universal projection, given explicitly by

πn(ada⋯dan) = adKa ∧⋯ ∧ dKan πn(da⋯dan) = dKa ∧⋯ ∧ dKan

¿e cohomology of this dierential algebra is denoted by HdR(A), and is called the de Rham

cohomologie of the commutative unital algebra A. ¿is terminology comes from the fact that thisdierential calculus looks very much like the de Rham dierential calculus (see Example ..). ⧫

Example .. (Polynomial algebra)LetV be a nite dimensional vector space overC. Consider the commutative algebra SV of polyno-mials on V. ¿en one has the identication SV ⊗⋀nV = Ωn

SV∣C by the map a⊗ v ↦ adKv in degree. ¿is dierential calculus is the “restriction” of the de Rham dierential calculus of C∞ functionsto the subalgebra of polynomial functions. ⧫

Example .. (Spectral triplet)Let A be an involutive unital associative algebra. A spectral triplet on A is a triplet (A,H,D) whereH is a Hilbert space on which an involutive representation ρ of A is given, and D is a self-adjointoperator on H (not necessarily bounded), whose resolvant is compact, and such that [D,ρ(a)] isbounded for any a ∈ A. ¿e operator D is called a Dirac operator.

¿e map π ∶ ΩU(A)→ B(H) dened by π(adUa⋯dUan) = a[D,a]⋯[D,an] is an involutive

representation of ΩU(A) onH.


Dene J = ⊕n≥(Kerπ ∩ΩnU(A)). One can show that J = J + dU J is a dierential two-sided

ideal in ΩU(A). ¿e dierential calculus dened by the spectral triplet (A,H,D) is the graded

dierential algebra ΩD(A) = Ω

U(A)/J.¿is construction is inspired by the denition of Fredholm modules, which are the building

blocks of K-homology (see Remark ..).See [Connes, ] and [Gracia-Bondía et al., ] for more details and examples. ⧫

Example .. (Derivations based dierential calculus for associative algebra)Let A be an associative algebra. ¿e space of derivations on A,

Der(A) = X ∶ A→ A / X linear map and X(ab) = (Xa)b + a(Xb)

is a Lie algebra and a module over the center Z(A) of A.Let Ωn

Der(A) be the set of Z(A)-multilinear antisymmetric maps Der(A)n → A. Dene onΩDer(A) =⊕n≥Ωn

Der(A) the product

(ωη)(X, . . . ,Xp+q) =p!q! ∑

σ∈Sp+q

(−)sign(σ)ω(Xσ(), . . . ,Xσ(p))η(Xσ(p+), . . . ,Xσ(p+q))

for any Xi ∈ Der(A), any ω ∈ ΩpDer(A) and any η ∈ Ωq

Der(A). Introduce on this graded algebra thedierential d ∶ Ωn

Der(A)→ Ωn+Der(A):

dω(X, . . . ,Xn+) =n+∑i=

(−)i+Xiω(X, . . .i∨. . . . ,Xn+)

+ ∑≤i<j≤n+

(−)i+jω([Xi,Xj], . . .i∨. . . .

j∨. . . . ,Xn+)

¿en (ΩDer(A), d) is a dierential calculus on A.

¿is dierential calculus is not a priori generated by A in degree . ¿e dierential calculusgenerated by A in (Ω

Der(A), d) is denoted by (ΩDer(A), d).

For A = C∞(M), the Lie algebra Der(A) is the Lie algebra of vector elds on the manifold M,and this dierential calculus (the two coincide here) is the de Rham dierential calculus.

For A = Mn(C), the Lie algebra Der(A) identies with sln(C), and the dierential calculusidenties with the Lie complex Mn(C) ⊗ ⋀sln(C)∗ for the adjoint representation of sln(C) onMn(C).

See [Dubois-Violette, ], [Dubois-Violette et al., b], [Dubois-Violette et al., a], [Dubois-Violette andMasson, ], [Masson, ], [Masson and Sérié, ] formore details, examples andapplications. ⧫

1.4.2 Hochschild homology¿eHochschild homology will not be presented here in its full generality. We refer to [Pierce, ],[Loday, ] and [Gerstenhaber and Schack, ] (for instance) to get further developments. Whatwill be presented here is the relation between Hochschild homology with values in the algebra itselfand the dierential calculi introduced above. ¿ese constructions are necessary to introduce andunderstand cyclic homology.

Let A be an associative algebra, not necessarily unital. As usual we denote by A+ = C ⊕ A itsunitarization.


¿eHochschild homologywe are interested in is dened using the following bicomplex, denotedby CC()

, (A), with only two non zero columns:

⋮b

⋮−b′

A⊗n+

b

A⊗n+−too

−b′

A⊗n

b

A⊗n−too

−b′

⋮b

⋮−b′

A A−too

t ∶ A⊗n → A⊗n is the cyclic operator:

t(a ⊗⋯⊗ an) = (−)n+an ⊗ a ⊗⋯⊗ an−

b ∶ A⊗n+ → A⊗n is the Hochschild boundary for the Hochschild complexwith values in A:

b(a ⊗ ⋅ ⋅ ⋅ ⊗ an) =n−∑p=

(−)pa ⊗ ⋅ ⋅ ⋅ ⊗ apap+ ⊗ ⋅ ⋅ ⋅ ⊗ an

+ (−)nana ⊗ a ⊗ ⋅ ⋅ ⋅ ⊗ an−

b′ ∶ A⊗n+ → A⊗n is the rst part of the Hochschild boundary b:

b′(a ⊗ ⋅ ⋅ ⋅ ⊗ an) =n−∑p=

(−)pa ⊗ ⋅ ⋅ ⋅ ⊗ apap+ ⊗ ⋅ ⋅ ⋅ ⊗ an

One can show the relations

b = b′ = b( − t) = ( − t)b′

¿e total complex of this bicomplex is given in degree n by CC()n (A) = A⊗n+ ⊕ A⊗n, with the

total dierential

bH = (b − t −b′ )

in matrix form.Now, notice that CC()

n (A) = A⊗n+⊕A⊗n = A+⊗A⊗n = Ωn(A) in degree n ≥ and CC() (A) =

A. In this identication, the dierential bH takes the very simple expression

bH(ωda) = (−)n[ω,a]

Denition .. (Hochschild homology with values in the algebra)Let A be an associative algebra. ¿e Hochschild homology HH(A) is the homology of the totalcomplex of the bicomplexCC()

, (A) dened above, i.e. the homology of the complex (Ω(A),bH).⧫

¿is second complex takes the form

Ω(A) ⋯bHoo Ωn(A)bHoo Ωn+(A)bHoo ⋯bHoo

Notice that bH is of degree −, but the dierential, which has not appeared in this construction, is ofdegree .

Remark .. (¿e unital case)When the algebra is unital, the second column (the one with b′) is exact: it admits the homotopy

s(a ⊗⋯⊗ an) = 1⊗ a ⊗⋯⊗ an (..)


Using standard spectral sequence arguments on this bicomplex, the homology of the total complexis then the homology of the rst column. In this case, one recovers the denition of the Hochschildcomplex which is usually given in textbooks:

A ⋯boo A⊗nboo A⊗n+boo ⋯boo (..)

One can even go a step further. It is possible to consider a quotient of this complex, called thenormalized complex, and to show, by standard arguments coming from the theory of simplicialmodules, that its homology is the same as the homology of the previous complex.

¿is normalized complex is dened by removing any contributions coming from elements pro-portional to the unit in the last n factors in A⊗n+. ¿e rst factor is not aected because it is infact the A-bimodule in which the Hochschild homology takes its values. ¿e normalized complexis then dened on the spaces A⊗A

⊗n(where as before A = A/C1) on which it is easy to check that

the dierential b is well-dened. But now, one has the identication ΩnU(A) = A ⊗ A

⊗n, so that in

the unital case, the Hochschild homology can be computed from the complex

ΩU(A) ⋯boo Ωn

U(A)boo Ωn+U (A)boo ⋯boo (..)

⧫

Denition .. (¿e trace map)¿e trace map Tr ∶ CC()

n (Mn(A))→ CC()n (A) is the morphism of complexes dened by

Tr(α ⊗⋯⊗ αn) = ∑(i,...,in)

a,ii ⊗ ⋅ ⋅ ⋅ ⊗ an,ini

where αr = (ar,i j)i,j ∈Mn(A). ⧫

Proposition .. (Morita invariance of Hochschild homology)For any unital algebra A and any integer n, the trace map induces an isomorphism

HH(Mn(A)) ≃ HH(A)

In fact, Morita invariance of the Hochschild homology of unital algebras is stronger than the onepresented here. It is invariant for the Morita equivalence which we now dene:

Denition .. (Morita equivalence of algebras)One says that two algebras A and B are Morita equivalent if there exist an A-B-module M and aB-A-module N such that A ≃ M⊗B N and B ≃ N ⊗A M as bimodules over A and B respectively. ⧫

For instance, A is Morita equivalent to B = Mn(A) using M = An written as a row and N = An

written as a column.Morita invariance of the Hochschild homology can be extended to the class ofH-unital algebras,

which contains the unital algebras.

Denition .. (H-unital algebras)A H-unital algebra is an algebra A for which the complex (A⊗,b′) has trivial homology. ⧫


Example .. (¿e algebra C)In the case A = C, one has

HH(C) = C HHn(C) = 0 for n ≥ ⧫

Example .. (Tensor algebra)Let V be a nite dimensional vector space and A = T V the tensor algebra over V. Denote by t thecyclic permutation acting on A in each degree (the t dening the bicomplex). ¿en

HH(A) = ⊕m≥ (V⊗m/Ran( − t)) co-invariants under the action of t

HH(A) = ⊕m≥ (V⊗m)t invariants under the action of tHHn(A) = 0 for n ≥ ⧫

Example .. (Relation with Lie algebra homology)Any associative algebra A gives rise to a Lie algebra ALie where the vector space is A and the Liebracket is the commutator: [a,b] = ab − ba. In the following, A is supposed to be unital.

¿e permutation groupSn acts on A⊗n by σ(a ⊗⋯⊗ an) = aσ−() ⊗⋯⊗ aσ−(n). Let us deneєn = ∑σ∈Sn(−)sign(σ)σ ∶ A⊗n → A⊗n the total antisymmetrisation. It induces a natural morphism

єn ∶ ⋀nA→ A⊗n

a ∧⋯ ∧ an ↦ єn(a ⊗⋯⊗ an)

which can be shown to commute with the boundary ∂ of the Lie algebra complex ⋀ALie and theboundary bof theHochschild complex, so that themorphismof dierential complexes є ∶ (⋀ALie,∂)→(A⊗,b) induces a morphism in homologies

є♯ ∶ H(⋀ALie,∂)→ HH(A)

If one consider the universal enveloping algebra U(g) of a nite dimensional Lie algebra g, onecan show thatHH(U(g)) ≃ H(g;U(g))where on the right it is the ordinary Lie algebra homologydened with the complex (⋀g,∂). ⧫

Example .. (¿e commutative case)Let us suppose that A is a commutative unital algebra.

Consider the constructions of Example ... Here the Lie structure on ALie is trivial, so that∂ = , and the morphism є♯ is in fact a morphism є♯ ∶ ⋀A→ HH(A).

One can show that there is a natural map

⋀nA→ ΩnA∣C

a ∧⋯ ∧ an ↦ da ∧⋯ ∧ dan

through which є♯ factors. One then get a natural map (also denoted by є♯):

є♯ ∶ ΩA∣C → HH(A)

da ∧⋯ ∧ dan ↦ [єn(a ⊗⋯⊗ an)]

where on the right hand side the brackets mean the homology class. ⧫


Example .. (Polynomial algebra)For a nite dimensional vector space V and the commutative unital algebra SV of polynomials onV, one has

HH(SV) = SV ⊗⋀V = ΩSV∣C ⧫

¿is is a particular situation of a more general theorem for which we need the following deni-tion:Denition .. (Smooth algebras)A commutative algebraA is a smooth algebra if for any algebraB and any ideal I inB such that I = ,the map HomC(A,B) → HomC(A,B/I) is surjective. ¿is means that every morphism of algebrasA→ B/I can be li ed to a morphism of algebras A→ B. ⧫

¿en one has the following result:

¿eorem .. (Hochschild-Kostant-Rosenberg)For any unital smooth commutative algebra A, the map є♯ ∶ Ω

A∣C → HH(A) of Example .. is anisomorphism of graded commutative algebras:

HH(A) ≃ ΩA∣C

¿e natural map which identies a dierential forms in Ωn(A) to a dierential form in ΩnA∣C is

explicitly given by ada⋯dan ↦ n!adKa ∧⋯ ∧ dKan. Notice the extra factor

n! compared to theuniversal projection πn of Example ... ¿is factor is required to get a further identication of thedierential on the Kähler dierential calculus with the B operator in cyclic homology (see [Loday,]) and to get a morphism of graded commutative algebras.

Remark .. (Extension to topological algebras)One can generalize the denition of theHochschild homology given above to take into account sometopological structure on the algebraA. In order to do that, one denes the spacesA⊗n using a tensorproduct adapted to the topological structure on the algebra. ¿e Hochschild homology one obtainsin this way is called the continuous Hochschild homology.

For Fréchet algebras, such a continuous homology is well dened and leads to the next two veryinteresting examples. ⧫

Example .. (¿e Fréchet algebra C∞(M))Let M be a C∞ nite dimensional locally compact manifold. ¿en Connes computed its continu-ous Hochschild homology in [Connes, ] and found the following result which generalizes theHochschild-Kostant-Rosenberg theorem:

HHCont (C∞(M)) = Ω

C(M) (complexied de Rham forms)

For reasons that will be explained later, this isomorphism between vector spaces, which we denoteby ϕ, is explicitly given in terms of universal forms by

Ωk(C∞(M))→ ΩkC (M) Ωk+(C∞(M))→ Ωk+

C (M)

ω↦ ( iπ

)k (k)!πk(ω) ω↦ ( i

π)k+

(k + )!πk+(ω)

where π ∶ Ω(C∞(M))→ ΩC(M) is the universal map dened in Example ... ⧫


Example .. (¿e irrational rotation algebra)¿e continuous Hochschild homology of the Fréchet algebra A∞θ (θ irrational) has been computedby Connes in [Connes, ]. Let λ = exp(iπθ).

If λ satises some diophantine condition (there exists an integer k such that ∣ − λn∣− is O(nk)),then

HHCont (A∞θ ) = C HHCont

(A∞θ ) = C

For any λ:

HHCont (A∞θ ) = C HHCont

n (A∞θ ) = 0 for n ≥

If λ does not satisfy some diophantine condition, HHCont (A∞θ ) and HHCont

(A∞θ ) are innite di-mensional. ⧫

Denition .. (Hochschild cohomology)Recall that the dual A∗ of an algebra A is a bimodule on A for the denition (aϕb)(c) = ϕ(bca) forany ϕ ∈ A∗ and a,b, c ∈ A. ¿e Hochschild complex (C(A),δ) for the Hochschild cohomologywith values in the bimodule A∗ is dened by

Cn(A) = Hom(A⊗n,A∗) = Hom(A⊗n+,C)

and by

δϕ(a ⊗ a ⊗⋯⊗ an+) =n

∑p=

(−)pϕ(a ⊗ a ⊗⋯⊗ apap+ ⊗⋯⊗ an+)

+ (−)n+ϕ(an+a ⊗ a ⊗⋯⊗ an)

By construction, δ is the adjoint to b in homology.¿e cohomology of this complex is denoted by HH(A). ⧫

1.4.3 Cyclic homology

Cyclic homology is dened using a bicomplex CC,(A) constructed using the bicomplex CC(), (A)

of the Hochschild homology. In order to do that, we need a new operator.N ∶ A⊗n → A⊗n is the norm operator dened by

N = + t +⋯ + tn

¿en one has the relations

( − t)N = N( − t) = b′N = Nb

¿e bicomplex CC,(A) is a repetition of the bicomplex CC(), (A) innitely on the right, using N


to connect them. In terms of the algebra A, one has

⋮b

⋮−b′

⋮b

⋮−b′

A⊗n+

b

A⊗n+−too

−b′

A⊗n+Noo

b

A⊗n+−too

−b′

⋯Noo

A⊗n

b

A⊗n−too

−b′

A⊗nNoo

b

A⊗n−too

−b′

⋯Noo

⋮b

⋮−b′

⋮b

⋮−b′

A A−too ANoo A−too ⋯Noo

Denition .. (Cyclic homology)Let A be an associative algebra. ¿e cyclic homology HC(A) of A is the homology of the totalcomplex of the bicomplex CC,(A) dened above. ⧫

Any morphism of algebras φ ∶ A → B induces a natural map CC,(A) → CC,(B) of bicom-plexes, so that one gets an induced map in cyclic homology φ♯ ∶ HC(A)→ HC(B).Remark .. (¿e Connes complex)In [Connes, ], Connes introduced cyclic cohomology, a dual version of cyclic homology. ¿eway he introduced it did not rely on a bicomplex, but on a subcomplex of the Hochschild complexfor cohomology. Some details of this construction are given in Example ... In a dual version,one can dene the Connes complex to compute cyclic homology as a quotient of the Hochschildcomplex for homology for a unital algebra.

To the bicomplex dened above, add a column on the le whose spaces are the cokernels of themorphisms ( − t) ∶ A⊗n+ → A⊗n+, which we denote by Cλ

n(A) = A⊗n+/Ran( − t). One can thencheck that the operator b is a well-dened operator on Cλ

(A) = ⊕n≥Cλn(A) which satises b = .

Denote byHλ(A) the homology of this complex. ¿e total complex TCC(A) of CC,(A) projects

onto the complex Cλ(A), sending the column p= onto Cλ

(A) and the other columns onto 0. Onethen gets a morphism in homology

HC(A)→ Hλ(A)

When the eld over which the algebra is dened containsQ, this is an isomorphism. To show that,one introduces an explicit homotopy for the horizontal operators which shows that the horizontalhomology of CC,(A) is trivial. By standard arguments on bicomplexes, this proves the assertion.⧫

Remark .. (¿e horizontal homology of CC,(A))One can show that for any algebra A, the homology of any row of CC,(A) is the group homologyH(Cn+;A⊗n+) of the cyclic groupCn+with values in theCn+-moduleA⊗n+ (for the action inducedby t). ⧫

We have seen that the total complex of CC(), (A) can be written in terms of the universal dif-

ferential calculus Ω(A) with the boundary operator bH. We can do something similar here. Everygrouping of two columns isomorphic toCC()

, (A) can be “compressed” as we did for theHochschild


bicomplex. ¿e operators b,b′ and ( − t) are then replaced by the unique operator bH ∶ Ωn(A) →Ωn−(A). ¿e operator N is replaced by a new operator B ∶ Ωn(A) → Ωn+(A), which takes thematrix form

B = ( N )

in the decomposition Ωn(A) = A⊗n+ ⊕ A⊗n. In order to have a pleasant diagram representing thenew bicomplex, li vertically each column on the right in proportion to its degree in the horizontaldirection. We then get the following (triangular) bicomplex

⋮bH

⋮bH

⋮bH

⋮bH

Ωn+(A)bH

Ωn(A)Boo

bH

⋯Boo Ω(A)bH

Boo Ω(A)Boo

Ωn(A)bH

Ωn−(A)Boo

bH

⋯Boo Ω(A)Boo

⋮bH

⋮bH

⋮bH

Ω(A)bH

Ω(A)bH

Boo Ω(A)Boo

Ω(A)bH

Ω(A)Boo

Ω(A)¿e total homology of this bicomplex is again the cyclic homology of A.

Denition .. (Mixed bicomplex)Amixed bicomplex is aN-graded complexM =⊕n≥Mn equipped with a dierential bM of degree− and a dierential BM of degree + such that

bMBM + BMbM =

¿e homology of the complex (M,bM) is called the Hochschild homology of the mixed bicom-plex, and it is denoted by HHM = H(M,bM).

We associate to such a mixed bicomplex the N-graded complex M dened by

Mn =⊕p≥

Mn−p

on which we introduce the dierential operator B′M − bM where B′M ∶ Mn → Mn− is BM on Mn−psuch that < p≤ n and onMn. ¿e cyclic homology of the mixed bicomplex is the homology ofthis dierential complex: HCM = H(M,B′M − bM)

Two mixed bicomplexes (M,bM,BM) and (N,bN ,BN) are said to be b-quasi-isomorphic ifthere exists a morphism of mixed bicomplexes φ ∶ (M,bM,BM) → (N,bN ,BN) (φ is of degree and commutes with the b’s and B’s) which induces an isomorphism in Hochschild homology. ⧫


Proposition ..Two b-quasi-isomorphic mixed bicomplexes have the same cyclic homology.

Example .. (¿e mixed bicomplex (Ω(A),bH,B))¿e motivation for the above denition is the example of the N-graded module Ω(A) with twodierentials bH and B. ¿e denitions of Hochschild and cyclic homology reproduce the ones weintroduced before. ⧫

Example .. (¿e mixed bicomplex (HH(A),,B♯))Because the dierentialB commuteswith the dierential bH, it denes amorphismB♯ on theHochschildhomologyHH(A). With this inducedmorphism, the triplet (HH(A),,B♯) is amixed bicomplex,whose Hochschild homology is the Hochschild homology of A. Indeed, when taking Hochschildhomology, bH is mapped to the zero operator.

Now, using standard argument on the spectral sequence constructed on the ltration by verticaldegree, one can see that this mixed complex computes the cyclic homology of A. ⧫

Remark .. (Some ideas to compute cyclic homology)Example .. tells us that in order to compute cyclic homology, one can rst compute Hochschildhomology, then look at the operatorB♯ induced byB, and compute theB♯-homology. Many examplesof concrete computations of cyclic homology are performed this way. Obviously, this supposes thatHochschild homology is computable!

Another approach is to consider the simplest possible dierential complex which computes theHochschild homology of the algebra, and then guess a B operator on it in order to build a mixed bi-complex b-quasi-isomorphic to one of the standardmixed bicomplexes givenhere. BecauseHochschildhomology can be dened through projective resolutions, such simple dierential complexes are usu-ally possible to nd. ⧫

Example .. (Mixed bicomplexes for the unital case)Let us suppose now that the algebra A is unital. ¿en we know that the Hochschild homology canbe computed with the complex (..). Using the ideas of Remark .., one can nd a B operatoron this complex in order to make it into a mixed bicomplex.

¿is operator is dened by B = ( − t)sN ∶ A⊗n → A⊗(n+) where t and N have been introducedbefore, and s ∶ A⊗n → A⊗(n+) is the homotopy (..). Explicitly, one has

B(a ⊗ a ⊗⋯⊗ an) =n−∑p=

[(−)np1⊗ ap⊗⋯⊗ an ⊗ a ⊗⋯⊗ ap−

− (−)n(p−)ap− ⊗ 1⊗ ap⊗⋯⊗ an ⊗ a ⊗⋯⊗ ap−]

In low degrees, these expressions take the following forms

B(a) = 1⊗ a + a ⊗ 1B(a ⊗ a) = (1⊗ a ⊗ a − 1⊗ a ⊗ a) + (a ⊗ 1⊗ a − a ⊗ 1⊗ a)


¿is mixed bicomplex (A⊗(+),b,B) is represented by the diagram

⋮b

⋮b

⋮b

A⊗

b

A⊗

b

Boo ABoo

A⊗

b

ABoo

ANow, one can perform this procedure with the complex (..). ¿en one obtains the same oper-

ator B and themixed bicomplex (ΩU(A),b,B)which is b-quasi-isomorphic to themixed bicomplex

of Example .. through the natural projection Ω(A)→ ΩU(A). ⧫

Example .. (¿e commutative case)Let us consider the notations and results of Example .., where the algebra is over the eldC. Onecan introduce the mixed bicomplex (Ω

A∣C,,dK) based on the Kähler dierential calculus, whichtakes the diagrammatic form

⋮

⋮

⋮

ΩA∣C

ΩA∣C

dKoo ΩA∣C

dKoo

ΩA∣C

ΩA∣C

dKoo

ΩA∣C

One can show that there is a naturalmorphismofmixed bicomplexes (ΩU(A),b,B)→ (Ω

A∣C,,dK),so that there is a natural map

HCn(A)→ ΩnA∣C/dKΩn−

A∣C ⊕Hn−dR (A)⊕Hn−

dR (A)⊕⋯

the last term being HdR(A) or H

dR(A) depending on the parity of n.Using¿eorem .., for any smooth unital commutative algebra A, one has the isomorphism

HC(A) ≃ ΩA∣C/dKΩ−

A∣C ⊕H−dR (A)⊕H−

dR (A)⊕⋯

In particular, this is the case for the polynomial algebra A = SV over a nite dimensional vectorspace V. ⧫

¿ere are a lot of structural properties on the cyclic homology groupswhich help a lot to computethem. We refer to [Loday, ] to explore them. Let us just mention the following result:


Proposition .. (Connes long exact sequence)¿ere are morphisms I and S which induce the following long exact sequence

⋯ //HHn(A) I //HCn(A) S //HCn−(A) B //HHn−(A) I //⋯

In low degrees, one gets

⋯ //HH(A) I //HC(A) S //HC(A) B //HH(A) I //HC(A) S //0

and the isomorphism0 B //HH(A) I //HC(A) S //0

¿is long exact sequence is a direct consequence of the fact that the bicomplex CC(), (A) is

included as pairs of columns in the bicomplex CC,(A). ¿is inclusion gives rise to the short exactsequence of bicomplexes

0 //CC(), (A) I //CC,(A) S //CC−,(A) //0

which denes I and S. In homology this short exact sequence produces Connes long exact sequence.¿e morphism S is called the periodic morphism.

Example .. (HC(C))Using the results of Example .. and the mixed bicomplex of Example .., one easily gets

HCn(C) = C HCn+(C) = 0

Using Connes long exact sequence, one has an isomorphism S ∶ HCn(C) → HCn−(C). Denote byun ∈ HCn(C) = C the canonical generator. ¿en, one can show that there is an isomorphism ofcoalgebras HC(C) ≃Ð→ C[u] explicitly given by un ↦ un, where the coproduct on C[u] is ∆(un) =∑p+q=n up⊗ uq.

For any algebra A, HC(A) is a comodule over the coalgebra HC(C):

HC(A)→ HC(A)⊗C[u]x ↦∑

p≥Sp(x)⊗ up

where Sp is the p-th iteration of S. ¿is is a concrete interpretation of the periodic morphism S. ⧫


Let us now dene the periodic and negative cyclic homologies. In order to do that, let us intro-duce the bicomplex CCper

,(A), innite in the two horizontal directions:

⋮b

⋮−b′

⋮b

⋮−b′

⋯ A⊗n+

b

Noo A⊗n+−too

−b′

A⊗n+Noo

b

A⊗n+−too

−b′

⋯Noo

⋯ A⊗n

b

Noo A⊗n−too

−b′

A⊗nNoo

b

A⊗n−too

−b′

⋯Noo

⋮b

⋮−b′

⋮b

⋮−b′

⋯ ANoo A−too ANoo A−too ⋯Noo

p= − − ⋯¿e bicomplex CC,(A) is naturally included in CCper

,(A) as the sub-bicomplex for which p ≥ .Denote by CC−,(A) the sub-bicomplex dened by p≤ .Denition .. (Periodic and negative cyclic homology)¿e periodic cyclic homology HP(A) of A is the homology of the total complex (for the product)dened from CCper

,(A) byTCCper

n (A) = ∏p+q=n

CCperp,q(A)

for any n ∈ Z.¿e negative cyclic homology HC− (A) of A is the homology of the total complex (for the prod-

uct) dened from CC−,(A) by

TCC−n(A) = ∏p+q=n(p≤)

CC−p,q(A) ⧫

Let us recall that in this situation, as the two bicomplexes we consider are innite in the le direction, direct sum and product do not coincide. An element in the direct sum contains only anite number of non zero elements in the spaces CCper

p,q(A) for p+ q = n, but an element in theproduct can be non zero in all of these spaces. If we were using direct sums to dene their totalcomplexes, then it would be possible to show that the associated homologies were trivial if the baseeld containsQ.

Using an adaptation of the procedure described for the cyclic homology, one can dene the cyclicperiodic homology of a mixed bicomplex, as well as its cyclic negative homology. ¿en one has:

Proposition ..Two b-quasi-isomorphicmixed bicomplexes have the same cyclic periodic homology and the same cyclicnegative homology.

¿e natural inclusion and the natural projection

I ∶ CC−,(A)→ CCper,(A) p ∶ CCper

,(A)→ CC,(A)


induce morphisms in homology

I ∶ HC−n(A)→ HPn(A) p ∶ HPn(A)→ HCn(A)

Proposition .. (-periodicity of HP(A))¿e periodic map S dened on the periodic bicomplex by translating on the le through two columns isan isomorphism. It induces the natural -periodicity:

HPn(A) ≃ HPn−(A)

which means that HP(A) is Z-graded.

From now on, we will use the notation HPν(A), with ν = , .We saw a Z-graded situation earlier for complex K-theory. Here is another similitude proved

in [Cuntz and Quillen, ]:

Proposition .. (Six term exact sequence)For any short exact sequence of associative algebras 0 //I //A //A/I //0 , one has the sixterm exact sequence

HP(I) // HP(A) // HP(A/I)δ

HP(A/I)δ

OO

HP(A)oo HP(I)oo

Proposition .. (Morita invariance)For any integer n ≥ , the tracemapdened inDenition .. induces an isomorphismTr♯ ∶ HPν(Mn(A)) ≃Ð→HPν(A).

Notice that we did not mention such a result for cyclic homology, because it is not true! ¿ere isa Morita invariance for cyclic homology on H-unital algebras (see Denition ..), but not on allalgebras.

Example .. (HPν(C) and HC−(C))

One has

HP(C) = C HP(C) = 0

¿ere is an isomorphism of algebrasHC− (C) ≃ C[v] for a generator v ∈ HC−−(C). ¿e productby v corresponds to the operation S ∶ HC−n(C)→ HC−n−(C). ⧫

Example .. (Tensor algebra)Let us use the notations of Example ... ¿e inclusion C → T V induces an isomorphism inperiodic cyclic homology:

HP(T V) = C HP(T V) = 0 ⧫

Example .. (¿e Laurent polynomials)Let A = C[z,z−] be the Laurent polynomials for the variable z. ¿en one has

HP(C[z,z−]) = C HP(C[z,z−]) = C ⧫


Example .. (Unital smooth commutative algebras)For any unital smooth commutative algebra A, one has

HP(A) = HevendR (A) =∏

p≥HpdR(A) HP(A) = Hodd

dR (A) =∏p≥Hp+dR (A)

If we compare this result with the cyclic homology groups given at the end of Example .., onesees that periodic cyclic homology is not ill with the edge eects on the le of the cyclic bicomplexwhich produce the contributions Ω

A∣C/dKΩ−A∣C. ⧫

Remark .. (Extension to topological algebras)As for Hochschild homology, one can generalize the denitions given above to take into accountsome topological structure on the algebraA, by replacing the tensor products with topological tensorproducts. We then obtain “continuous” versions of these cyclic homologies.

In [Cuntz, ], Cuntz proved a six term exact sequence as in Proposition .. for a restrictedclass of topological algebras, calledm-algebras (see also [Cuntz et al., ]). ⧫

Example .. (Continuous cyclic homology of Banach algebras)On Banach algebras, the continuous cyclic homologies are not interesting. For instance, for com-mutative C∗-algebras, one gets

HPcont (C(X)) = bounded measures on X HPcont (C(X)) = 0 ⧫

¿is example shows that cyclic homology is not a very powerful theory for noncommutativetopological spaces. ¿e following result conrms this fact. We need a

Denition .. (Dieotopic morphisms)Let A and B be two associative algebras. Two morphisms of algebras φ,φ ∶ A → B are said to bedieotopic if there exists a morphism of algebras φ ∶ A→ B⊗C∞([, ]) such that φt coincides withφ (resp. φ) when evaluated at t = (resp. at t = ) in the target algebra. Notice that the tensorproduct B⊗ C∞([, ]) is purely algebraic. ⧫

Proposition .. (Dieotopic invariance)If φ and φ are dieotopic then they induce the same morphism HPν(A)→ HPν(B).

¿ere is no general homotopic invariance result on periodic cyclic homology.If you need a result more to convince you that cyclic homology is well adapted to dierential

structures, here is the main result, obtained by Connes in [Connes, ], which is a generalisationof Example ..:

Example .. (Continuous periodic cyclic homology of C∞(M))LetM be a C∞ nite dimensional locally compact manifold. ¿en one has

HPcont (C∞(M)) = HevendR (M) HPcont (C∞(M)) = Hodd

dR (M) ⧫

Remark .. (Comparing cyclic homology with K-theory)In the next section, we will establish a very strong relation between K-theory and periodic cyclichomology. Z-graduation, Morita invariance and the six term exact sequence give us obvious simi-larities between these two theories.


But, we would like to make it clear that the two theories are very dierent on an essential point.We notice in Remark .. that K-theory can be computed using some dense subalgebra (stable byholomorphic functional calculus). ¿e situation is clearly not the same for periodic cyclic homology:compare Example .. with Example ...

K-theory is an homology theory for noncommutative topological spaces. Periodic cyclic homol-ogy is an homology theory for algebras concealing some dierentiable properties. ⧫

Example .. (First version of cyclic cohomology)We dene here the cyclic cohomology using the dierential complex (Cλ(A),δ), which is the rstversion of cyclic cohomology exposed in [Connes, ].

By denition,

Cnλ(A) = ϕ ∈ Hom(A⊗n+,C) / ϕ(a ⊗⋯⊗ an ⊗ a) = (−)nϕ(a ⊗ a ⊗⋯⊗ an)

⊂ Cn(A)

and δ is the Hochschild boundary operator for the complex C(A) restricted to this subspace (seeDenition ..). Indeed, the main remark made by Connes to dene its cyclic complex as a sub-complex of a Hochschild complex was that the cyclic condition dening the ϕ’s in Cn(A) which areelements of Cn

λ(A) is compatible with the boundary δ.¿e cohomology of the complex (Cλ(A),δ) is the cyclic cohomology HC(A) of A.For n = , a cycle ϕ is a trace on A, because (δϕ)(a ⊗ a) = ϕ(aa) − ϕ(aa) = . ¿e cyclic

complex of Connes is explicitly dened to generalize this property to higher degrees. So, cycliccohomology is a theory of generalized traces.

On this cohomology, the inclusion Cλ(A) C(A) at the level of complexes induces a mapI ∶ HC(A)→ HH(A) and Connes long exact sequence

⋯ //HHn(A) B //HCn−(A) S //HCn+(A) I //HHn+(A) B //⋯

In this long exact sequence, the two maps B and S are not so easy to dene as in the previous con-struction for cyclic homology.

Nevertheless, one can show that the periodic map S ∶ HCn(A) → HCn+(A) can be used todene periodic cyclic cohomology as

HP(A) = limÐ→(HCn+(A),S) HP(A) = limÐ→(HCn(A),S)

which explains the name “periodic” for the cohomology group and the map S. ⧫

One denes the dual bicomplex CC,(A) of CC,(A) replacing in each bidegree (p,n) thespace A⊗n+ by the space Hom(A⊗n,A∗) = Hom(A⊗n+,C), and adjoining the four maps b, b′, t andN. Recall that the adjoint of b is the δmap introduced in Denition ...

In the same manner, one denes the bicomplex CC,per(A) from the bicomplex CCper,(A). With

these bicomplexes one can dene cyclic cohomology in its generality.

Denition .. (Cyclic cohomology)¿e cyclic cohomology HC(A) of A is the cohomology of the total complex of the bicomplexCC,(A).

¿e cyclic periodic cohomology HP(A) of A is the cohomology of the total complex of thebicomplex CC,per(A). Here, the total complex is constructed using direct sums. ⧫


As for periodic cyclic homology, HP(A) is Z-graded.

Remark .. (Entire cyclic cohomology)In the denition of HP(A), one uses the direct sum to construct the total complex. ¿is is thedual version of the direct product used for periodic cyclic homology. Indeed, one can show that thedirect product would produce a trivial cohomology. Using direct sum in periodic cyclic cohomologypermits one to dene a natural pairing with periodic cyclic homology: cochains in periodic cycliccohomology have nite support, so that only a nite number of terms are non zero when evaluatedon a (innite) chain in cyclic periodic homology.

Let A be a Banach algebra. ¿en one denes a norm on Hom(A⊗n+,C) by ∥ϕn∥ = sup∣ϕ(a ⊗⋯⊗ an)∣ / ∥ai∥ ≤ .

Denote by TCC∏(A) the total complex of CC,per(A) obtained using direct product. Each ele-ment in TCCp

∏(A) is an innite sequence (ϕn) or (ϕn+) according to parity of p. One denesa subcomplex ECC(A) of TCC∏(A) imposing a growing condition on such an innite sequence:the radius of convergence of the series∑n≥ ∥ϕn∥zn/n! (resp. ∑n≥ ∥ϕn+∥zn/n!) is innity.

¿e entire cyclic cohomology HE(A) is dened as the cohomology of the complex ECC(A).One can show that HE(A) is Z-graded, as is the periodic cyclic cohomology.

Any cochain dening an element in HP(A) has nite support, so that there is an natural mapHP(A) → HE(A). ¿is map is an isomorphism in some cases, for instance A = C, but not ingeneral. See [Khalkhali, ] for examples of such isomorphisms. ⧫

Example .. (¿e irrational rotation algebra)For irrational θ, one has

HPcont (A∞θ ) = C HPcont (A∞θ ) = HH(A∞θ )/RanB♯ = C

¿ere is no need to mention any diophantine condition here (see Example ..).In periodic cyclic cohomology, one of the two classes in HPνcont(A∞θ ) = C is Sτ where τ is the

unique normalised trace on A∞θ , τ(∑m,n∈Z am,nUmVn) = a,, and the second one is expressed interms of the continuous derivations δ and δ:

φ(a ⊗ a ⊗ a) =iπ

τ[a(δ(a)δ(a) − δ(a)δ(a))] ⧫

Remark .. (Pairing with K-theories)A Fredholm module (H,ρ,F) over the C∗-algebra A is called p-summable if [F,a] ∈ Lp(H) forany a ∈ A (we write a for ρ(a) from now on). ¿e space Lp(H) = T ∈ K / ∑∞

n= µn(T)p < ∞with µn(T) the n-th eigenvalue of ∣T∣ = (T∗T)/, is the Schatten class. It is a two-sided ideal in B,and a Banach space for the norm ∥T∥p = (∑∞

n= µn(T)p)/p = Tr(∣T∣p)/p. For any S ∈ Lp(H) andT ∈ Lq(H), one has ST ∈ Lr(H) for

r = p +

q and ∥ST∥r ≤ ∥S∥p∥T∥q.Let (H,ρ,F) be a p-summable Fredholm module (odd or even according to the parity of p− )

and denote by γ its grading map if it is even. For any operator T onH such that FT + TF ∈ L(H),let us dene Tr′(T) =

Tr(F(FT + TF)). For any n ≥ such that n+ ≥ p− in the odd case andn ≥ p− in the even case, and for any ai ∈ A, the expressions

φn+(a ⊗⋯⊗ an+) = Tr′(a[F,a]⋯[F,an+]) in the odd caseφn(a ⊗⋯⊗ an) = Tr′(γa[F,a]⋯[F,an]) in the even case


make sense and dene an odd or an even cocyle inHC(A), which depends only on theK-homologyclass of the Fredholm module. In fact, using correct normalizations, this denes a natural pairingHPν(A) × Kν(A)→ C for ν = , . See [Connes, ] and [Gracia-Bondía et al., ] for details.

In the next section, the Chern character will realize a pairing HPν(A) × Kν(A)→ C. ⧫

1.5 The not-missing link: the Chern character

¿e Chern character is a special characteristic class dened rst in the topological context. It wasused to related the K-theory of a topological space to its cohomology. When Connes introducedcyclic homology, he saw immediately that a purely algebraic generalisation was possible, which con-nects the K-theory for algebras and the periodic cyclic homology. Now, the Chern character is ex-tensively studied, because it helps interpret a lot of previous results in dierent areas of mathematics,which where not so well understood.

1.5.1 The Chern character in ordinary differential geometryLet us recall some basic facts about characteristic classes for vector bundles.

Let G be a topological group. ¿en one has:Proposition .. (Classifying space BG)¿ere exists a G-principal bre bundle EG → BG such that for any G-principal bre bundle P over atopological space X, there exists a continuous map fP ∶ P → BG such that P = f∗P EG (the pull-backbre bundle). BG is called the classifying space of the topological groupG and fP the classifying map ofP.

Recall that the pull-back P = f∗Q of a bre bundleQ → Y through a continuous map f ∶ X→ Yis dened by Px = Qf(x) for any x ∈ X. If g ∶ X → Y is homotopic to f then f∗Q and g∗Q areisomorphic.

One can show that EG is a contractible space, so that its homology is not very interesting. ¿eimportant object in this proposition is BG:Proposition .. (Classication ofG-principal bre bundles)¿e space of isomorphic classes of G-principal bre bundles over X is [X;BG], the space of homotopicclasses of continuous maps X→ BG.

¿is space is not easy to compute, so that this classication remains just an identication withoutany practical utility in general. ¿is leads us to consider other objects to try to classify G-principalbre bundles, in terms of cohomology classes:Denition .. (Characteristic classes)A characteristic class of P in a cohomology group H(X;A) with coecient in the abelian group A,is the pull-back by fP of any cohomology class c ∈ H(BG;A). ⧫

Characteristic classes depend upon the coecient groupA, which is o en essential tomake someconcrete interpretations of certain characteristic classes.

If one is interested in vector bundles instead of principal bundles, the previous construction canbe performed with the associated principal bundle. Any vector bundle is the pull-back through theclassifying map fP of a canonical vector bundle over BG. So that for any vector bundle E

πÐ→ X withstructure group G, one can introduce its characteristic classes as pull-back of classes in H(BG;A).We will use the notation c(E) for the pull-back of c ∈ H(BG;A) in H(X;A).


Proposition .. (Functoriality of characteristic classes)Let φ ∶ X→ Y be a continuous map, and E → Y a vector bundle on Y. ¿en for any characteristic classc one has c(φ∗E) = φ♯c(E) where φ♯ ∶ H(Y)→ H(X) is the ring morphism induced in cohomology.

Example .. (Discrete groups)In the case of a discrete group G, one can show that BG = K(G, ) is the Eilenberg-MacLane spaceof type (G, ), so that H(BG;Z) is the ordinary cohomology of groups H(G). ⧫

It is possible to construct explicitly the classifying spaces BG for a large class of groups. Here aresome examples.

Example .. (Some usual classifying spaces)

G Z Zn Z U() = S U(n) O(n)EG R Rn S∞BG S Tn RP∞ CP∞ G(n,C∞) G(n,R∞)

⧫

S∞ is the sphere in R∞, RP∞ = limÐ→RPn, CP∞ = limÐ→CPn, G(n,C∞) = limÐ→G(n,Cp) whereG(n,Cp) is the complex Grassmanian manifold. . .

Example .. (Cohomology groups of some classifying spaces)Here are some examples of cohomology groups of some classifying spaces.

We denote by A[a, . . . ,ap] the graded commutative ring generated over the abelian groups Aby the p elements ai (whose degrees will be given):

H(BU(n);Z) = Z[c, c, . . . , cn] H(BSU(n);Z) = Z[c, . . . , cn]

where deg ck = k. ¿e class ck is the k-th Chern class. ¿e class c = + c + c + ⋅ ⋅ ⋅ + cn is the totalChern class. It satises c(E⊕ E′) = c(E)c(E′) for any vector bundles E and E′.

H(BO(n);Z) = Z[p, p, . . . , p[n/]]

where deg pk = k and [n/] is the integer part of n/. ¿e class pk is the k-th Pontrjagin class. ¿eclass p= + p + p + ⋅ ⋅ ⋅ + p[n/] is the total Pontrjagin class which satises p(E⊕ E′) = p(E)p(E′).

H(BSO(m + );Z) = Z[p, p, . . . , pm] H(BSO(m);Z) = Z[p, p, . . . , pm−, e]

where deg pk = k and deg e = m. ¿e class e is called the Euler class, it satises e(E ⊕ E′) =e(E)e(E′).

H(BO(n);Z) = Z[w, . . . ,wn] H(BSO(n);Z) = Z[w, . . . ,wn]

where degwk = k is the k-th Stiefel-Whitney class. ¿e class w = + w + w + ⋅ ⋅ ⋅ + wn is the totalStiefel-Whitney class which satises w(E⊕ E′) = w(E)w(E′). ⧫

Example .. (Interpretation ofw andw)LetM be a locally compact nite dimensionalmanifold. M is orientable if and only if the rst Stiefel-Whitney class w(TM) of its tangent space TM is zero. IfM is orientable, it admits a spin structureif and only if the second Stiefel-Whitney class w(TM) is zero. ⧫

1.5 The not-missing link: the Chern character 55

Example .. (Classication of complex line vector bundles)¿e rst Chern class of c(L) ∈ H(X;Z) of a complex line vector bundle L→ X is a total invariantin the space of isomorphic classes of line vector bundles over X. ⧫

Example .. (Compact connected Lie groups)For any compact connected Lie group G, one has

Hn(BG;R) = PnI (g) Hn+(BG;R) = 0

where g is the Lie algebra of G and PI (g) is the graded algebra of invariant polynomials on g.For the compact Lie groups in Example .., these invariant polynomials are generated by the

formulas:det(λ1 + i

πX) = λn + c(X)λn− + c(X)λn− + ⋅ ⋅ ⋅ + cn(X)

for any X ∈ u(n);

det(λ1 − πX) = λn + p(X)λn− + p(X)λn− + ⋅ ⋅ ⋅ + pm(X)λn−m

for any X ∈ o(n);e(X) = (−)m

mπmm! ∑i,...,im

єii...im−imXii . . .Xim−im

for any X ∈ so(m), where єii...im−im is completely antisymmetric with є...m = . ¿e quantityPf(X) = (π)me(X) is called the Pfaan of X. It is a square root of the determinant. ¿e Euler classis then associated to a very particular invariant polynomial. ⧫

Example .. (Characteristic classes through connections)It is possible to construct characteristic classes directly using invariant polynomials in PI (g). Inorder to do that, consider a dierentiable principal bre bundle P→M over a dierential manifoldwith structure group G. Let us denote by ω ∈ Ω(P) ⊗ g a connection on P and Ω its curvature.Recall that ω is a covariant object for the action Rg ofG on P by right multiplication and the adjointaction Ad on g : R∗gω = Adg−ω for any g ∈ G. Its curvature is also covariant, R∗gΩ = Adg−Ω, andsatises the Bianchi identity dΩ+ [ω,Ω] = .

Let (U,ϕ) be a local trivialisation of P, where U is an open subset of M and ϕ ∶ U × G → P∣Uis a dieomorphism which intertwines the actions of G on P and G. Dene by sU(x) = ϕ(x, e) thesection which trivializes P∣U and by AU = s∗Uω and FU = s∗UΩ the local connection -form and thelocal curvature -form. If (V ,ψ) is a second trivialization of P, with U ∩ V ≠ ∅, then one has therelations

AV = g−UVAUgUV + g−UVdgUV FV = g−UVFUgUV

where gUV ∶ U ∩ V → G is the transition function between the two trivializations. ¿e local formsFU satisfy to a Bianchi identity.

Let us consider pan invariant polynomial on g, of degree k. ¿en one can dene p(FU , . . . ,FU)as a local k-form on U by evaluating p on the values of FU in g. Because p is invariant, one hasp(FU , . . . ,FU) = p(FV , . . . ,FV) so that it denes a global k-form onM. Using the Bianchi identity,one can then show that its dierential is zero. We then have associated to p a cohomology class inHk(M;R), which can be shown to be independent of the choice of the connection ω. ¿is mapPkI (g)→ Hk(M;R) is the Chern-Weil map.


¿is class is exactly the characteristic class given by the invariant polynomial p in the identica-tion Hn(BG;R) = PnI (g) in Example ...

One does not really need to express the connection -form and its curvature -form locally onan open set of the base space M. Indeed, p(Ω, . . . ,Ω) makes sense as a k-form on P. Using theproperties of the curvature -form Ω and the invariance of the polynomial p, one can show that itis a basic form for the action of G on P, and as such, it identies with a k-form on the base spaceM. ⧫Proposition .. (Decomposition principle)Let E, . . . ,Ep→ X be p complex vector bundles. ¿en there exist a manifold F and a continuous mapσ ∶ F → X such that the pull-backs σ∗Ei → F are all decomposed as direct sum of complex line vectorbundles, and such that the map induced in cohomology σ♯ ∶ H(X)→ H(F) is injective.

Why decompose a vector bundle in a direct sum of line vector bundles? ¿e answer is in thefollowing construction.

Let R(c(E), . . . , c(Ep)) be a polynomial relation in H(X) between the Chern classes of thevector bundles Ei. We would like to establish the relation R(c(E), . . . , c(Ep)) = for any vectorbundles over X, and for any X. Using the decomposition map σ ∶ F→ X and the functoriality of theChern classes (and the fact that the relation is a polynomial relation) we have

σ♯(R(c(E), . . . , c(Ep))) = R(c(σ∗E), . . . , c(σ∗Ep))

Now, let us assume that for any base spaceY and any vector bundles Fi overY which are direct sumofline vector bundles, the relation R(c(F), . . . , c(Fp)) = can be established. ¿en, for any Ei over X,the Fi = σ∗Ei over Y = F are direct sums of line vector bundles, so that the relation is true for them.¿e right hand side of the relation is then zero, which implies by injectivity of σ♯ ∶ H(X)→ H(F)that the relation is also zero for the Ei’s.

So, in order to establish an abstract relation between the Chern classes, it is sucient to show itfor any vector bundle decomposed as a direct sum of line vector bundles over any space.Example .. (Chern classes and elementary symmetric polynomials)Let us apply the relation c(E ⊕ E′) = c(E)c(E′), where c is the total Chern class, to a direct sum ofline vector bundles E = ℓ ⊕ ⋯ ⊕ ℓn. ¿en c(E) = c(ℓ)⋯c(ℓn). For a line vector bundle, one hasc(ℓ) = + c(ℓ). Denote by xi = c(ℓi) the rst Chern classes of these line vector bundles. ¿en onehas

c(E) =n

∏i=

( + xi) =n

∑j=σj(x, . . . ,xn)

where the functions σj are the elementary symmetric polynomials of total degree j. ¿ey are explic-itly given in terms of the n (commuting) variables Xi by

σ(X, . . . ,Xn) = σ(X, . . . ,Xn) = ∑≤i≤n

Xi σ(X, . . . ,Xn) = ∑≤i<j≤n

XiXj

⋯ σn(X, . . . ,Xn) = ∏≤i≤n

Xi

Any symmetric polynomial (resp. any formal symmetric series) in the n variables Xi can be ex-pressed as a polynomial (resp. a formal series) in these elementary symmetric polynomials:

C[X, . . . ,Xn]Sn = p ∈ C[X, . . . ,Xn] / p(X, . . . ,Xn) = p(Xσ−(), . . . ,Xσ−(n))= C[σ, . . . ,σn]


¿e previous computation shows us that the Chern classes can be written as cj(E) = σj(x, . . . ,xn)when E is decomposed. If E is not decomposed, then use σ∗E over F. ⧫

Example .. (Characteristic class associated to a symmetric polynomial)¿e previous Example gives us another application of the decomposition principle, which is to con-struct a new characteristic class in terms of the Chern classes, but writing it down explicitly onlyin terms of the rst Chern classes and a symmetric polynomial. Indeed, let p(X, . . . ,Xn) be asymmetric polynomials. ¿en it is a polynomial of the form R(σ, . . . ,σn). For any vector bun-dle E → X decomposed as E = ℓ ⊕ ⋯ ⊕ ℓn, dene the characteristic class cp(E) = p(x, . . . ,xn)where xi = c(ℓi). ¿en cp(E) = R(σ(x, . . . ,xn), . . . ,σn(x, . . . ,xn)) = R(c(E), . . . , cn(E)). Now,if E is not decomposed as a direct sum of line vector bundles, the last relation can be used to dene,without ambiguities, the class cp(E), thanks to the decomposition principle and the functoriality ofthe Chern classes.

¿is construction can be generalised to any invariant formal series in n variables. ⧫

Denition .. (¿e Chern character)Let E be a vector bundle over X. ¿e Chern character ch(E) of E is dened to be the characteristicclass associated to the formal series

p(x, . . . ,xn) = ex +⋯ + exn = n +n

∑i=xi +

n

∑i=

(xi) +⋯

Notice that the coecient group for the cohomology of this class is necessarilyQ, because the den-ing expression for ch(E)makes use of rational numbers. ⧫

Example .. (¿e invariant polynomial of the Chern character)We saw in Example .. that characteristic classes can be dened using a connection on the vectorbundle and an invariant polynomial. ¿e Chern character is a particular characteristic class, and itsinvariant polynomial (in fact an invariant formal series) is P(X) = Tr exp( i

πX), so that ch(E) =Tr exp ( iFπ) for any local curvature -form of a connection on E.

As a form on the principal bre bundle, this expression is

ch(ω) = Tr exp( iΩπ

) =∞∑k=

k!

( iπ

)k

Tr(Ωk) ⧫

Proposition .. (Product and additive properties of ch)Using the decomposition principle, one can show that for any vector bundles E and E′:

ch(E⊕ E′) = ch(E) + ch(E′) ch(E⊗ E′) = ch(E) ch(E′)

¿eorem .. (¿e Chern character as an isomorphism)¿e Chern Character denes a natural morphism of rings ch ∶ K(X) → Heven(X;Q) which inducesan isomorphism

ch ∶ K(X)⊗Z Q≃Ð→ Heven(X;Q)

for locally compact nite dimensional manifolds X. In that case, the Chern character can be extendedto a isomorphism ch ∶ K(X)⊗Z Q

≃Ð→ Hodd(X;Q).


Example .. (¿e Chern character K−(M)→ Hodd(M;Q))It is possible to give an expression of the Chern character in odd degrees using connections. Let ωandω be connections on the principal bre bundle P. ¿enωt = ω+ t(ω−ω) is also a connectionfor any t ∈ [, ]. We denote by Ωt its curvature. One can show that the Chern-Simons form

cs(ω,ω) = ∫

dtTr((ω − ω) exp( iΩt

π))

satises d cs(ω,ω) = ch(ω) − ch(ω) where ch(ω) is given as in Example ...Let g ∶ M → U(n) be a smooth map. Consider the trivial bre bundle P = M ×U(n), with the

two connections ω = and ω = g−dg. ¿en one denes

ch(g) = cs(, g−dg) =∞∑k=

(−)k k!(k + )! (

iπ

)k+

Tr((g−dg)k+)

¿is denes a map from the class of g in K−(M) into Hodd(M;Q). ⧫

1.5.2 Characteristic classes and Chern character in noncommutative geometryIt is possible to construct some characteristic classes, and in particular the Chern character, usingthe algebraic setting of modules and dierential calculi. ¿e construction of these classes are basedupon some generalisation of the construction of the Chern classes in terms of the curvature of someconnection. In order to do that, one need to dene the so-called noncommutative connections.

Let (Ω,d) be a dierential calculus on an associative unital algebra A, and let M be a niteprojective le module over A = Ω.

Denition .. (Noncommutative connection)A noncommutative connection on M for the dierential calculus (Ω,d) is linear map ∇ ∶ M →Ω ⊗A M such that ∇(am) = da⊗m + a(∇m) for anym ∈ M and a ∈ A. ⧫

Let us introduce M = Ω ⊗A M as a graded le module over Ω, and EndΩ(M) the gradedalgebra of Ω-linear endomorphisms on M (any T ∈ EndkΩ(M) satises T(η) ∈ Ωm+k ⊗A M andT(ωη) = (−)nkωT(η) for any η ∈ Ωm ⊗A M and any ω ∈ Ωn).

Let M′ be a le module such that M⊕M′ = AN and denote by p ∶ AN → M the projection andϕ ∶ M→ AN the inclusion. ¿en pϕ = IdM.

Using right multiplication on AN , one can make the identication EndΩ(AN) =MN(Ω).

Proposition .. (General properties of noncommutative connections)Any noncommutative connection ∇ can be extended into a map of degree + on M such that for anym ∈ M and ω ∈ Ωn,

∇(ω⊗A m) = (dω)⊗A m + (−)nω(∇m)

¿e space of connections is an ane space over EndΩ(M).

Denition .. (Curvature of a noncommutative connection)¿e curvature of ∇ is the map Θ = ∇ = ∇ ∇. ⧫

We dene the linear map δ ∶ EndΩ(M) → EndΩ(M) by the relation δ(T) = ∇T − (−)kT∇,where T ∈ EndkΩ(M). One can easily show that δ is a graded derivation of degree + on the gradedalgebra EndΩ(M).


Proposition ..One has Θ ∈ EndΩ(M), δ(T) = ΘT − TΘ = [Θ,T]gr and the Bianchi identity δ(Θ) = .

Example .. (Existence of connections)Let e ∈ EndA(AN) be a projector, and dene the le module M = e(AN). e is also a projector inEndΩ(Ω⊗AAN)which is naturally extended byΩ-linearity. ¿en, if∇ is a connection onAN , themapm↦ e(∇m) is a connection onM.

On AN there is a natural connection given by the dierential map of the dierential calculus:AN ∋ (a, . . . ,aN) ↦ (da, . . . ,dan) ∈ Ω ⊗A AN = (Ω)N . ¿en any nite projective module on Aadmits at least one connection. ⧫

Example .. (Direct sum of connections)Let (M,∇M) and (N,∇N) be two nite projective modules overA for the same dierential calculus.¿en ∇ ∶ M ⊕ N → Ω ⊗A (M ⊕ N) dened by ∇(m⊕ n) = (∇Mm) ⊕ (∇Nn) is a connection onM⊕ N which we denote by ∇M ⊕∇N . ⧫

Denition .. (Graded trace)Let V be a graded vector space. A graded trace on Ω with values in V is a linear morphism ofdegree , τ ∶ Ω → V, such that τ(ωη) = (−)mnτ(ηω) for any ω ∈ Ωm and η ∈ Ωn. ⧫

Notice that the restriction τ ∶ A = Ω → V is an ordinary trace on A.

Proposition .. (¿e universal trace)If we denote by [Ω,Ω]gr the subspace of Ω lineary generated by the graded commutators, then thegraded vector space Ω = Ω/[Ω,Ω]gr inherits the dierential of Ω, which we denote by d, and theprojection τΩ ∶ Ω → Ω is a graded trace which commutes which the dierentials.

For any graded trace τ ∶ Ω → V there is a factorisation τ = ττΩ for a morphism τ ∶ Ω → V.¿is is why τΩ is called the universal trace on Ω.

Example .. (¿e trace on EndΩ(M))Because of the identication EndΩ(AN) = MN(Ω), there is a natural trace on EndΩ(AN) withvalues in Ω induced by the trace on the matrix algebra MN(C), which we denote by Tr. For anyT ∈ EndΩ(M), one has T = ϕTp ∈ EndΩ(AN), so that we can dene τΩ(Tr(T)) ∈ Ω. ¿is map isa graded trace which does not depend upon p, ϕ and N. ¿e trace EndΩ(M)→ Ω will be denotedby TrΩ. It satises TrΩ δ = dTrΩ. ⧫

Denition .. (Characteristic classes ofM)For any integer k, the cohomology class of TrΩ(Θk) inHk(Ω, d) is independent of the connection∇. ¿is is the k-th characteristic class ofM for the dierential calculus (Ω,d). ⧫

Denition .. (¿e Chern character ofM)We dene the Chern character ch(M) ∈ Heven(Ω, d) associated toM by

chk(M) = [ (−)k(iπ)kk! TrΩ(Θ

k)] ∈ Hk(Ω, d)

ch(M) =∑k≥

chk(M) = TrΩ exp( iΘπ

) ∈ Heven(Ω, d) ⧫


Obviously, this denition is just an algebraic rephrasing of the expression that was given in Ex-ample ...

Example .. (¿e connection induced by a projector)We saw in Example .. that there is a natural connection onM = e(AN) expressed in terms of theprojector e ∈ EndA(AN). From now on, e will be identied with an element in the matrix algebraMN(Ω), which acts onAN bymultiplication on the right. One can compute explicitly the curvatureof this connection using this matrix algebra, and then one nds, for any a ∈ M ⊂ AN ,

Θ(a) = −a(de)(de)e

¿is expression can be used to express the Chern character ofM in terms of the matrix e :

ch(M) =∑k≥

[ (iπ)kk!τΩTr(e(de)

k)] ⧫

Proposition .. (Additive properties of ch)For any two nite projective le modulesM and N on A, one has

ch(M⊕ N) = ch(M) + ch(N)

Remark .. (No product property!)¿ere is no product property which could satisfy this Chern character, because there is no possibilityto dene a tensor product of two nite projective le modulesM and N. . .

Example .. (¿e geometric Chern Character)In the case A = C∞(M) and Ω = Ω(M), the de Rham dierential calculus, one has Ω = Ω

because Ω(M) is graded commutative. ¿en the Chern character takes its values in the evende Rham cohomology of M. By the Serre-Swan theorem in its algebraic version, ¿eorem ..,any nite projective module on C∞(M) is the space of smooth sections of a vector bundle E overM. It is easy to verify that a noncommutative connection is then an ordinary connection on E,seen as a covariant derivative maps on sections. ¿is identication uses the natural isomorphismΩ ⊗A M = Ω(M,E), where Ω(M,E) is the space of dierential forms on M with values in E.¿e curvature is then an element in Ω(M,End(E)) = Ω(M)⊗C∞(M) EndC∞(M)(Γ(E)), the spaceof -forms with values in the associated vector bundle End(E) = E ⊗ E∗. In this context, one hasEndΩ(Ω ⊗A M) = Ω(M,End(E)) and the trace is the ordinary trace on the bres of End(E).

Using these considerations and the explicit formulas dening them, the two denitions of theChern characters coincide.

As an exercise, one can show that the relation δ(Θ) = is indeed the Bianchi identity! ⧫

1.5.3 The Chern character from algebraic K-theory to periodic cyclic homology¿edenition of the (algebraic)Chern characterwewill use rests upon the two results concerning thealgebras C and C[z,z−] given in Examples .. and ..: HP(C) = C and HP(C[z,z−]) = C.Recall that the trace map Tr dened in Denition .. induces the Morita isomorphism in periodiccyclic homology.


LetA be an associative unital algebra. Let p ∈MN(A) be a projector. ¿en it denes a morphismof algebras ip ∶ C → MN(A) by λ↦ λp. Indeed, ∈ C is mapped to p, and the relation p = p is therequired compatibility with = . ¿is morphism is not a morphism of unital algebras.

Let u ∈MN(A) be an invertible element. ¿en it denes a morphism of algebras iu ∶ C[z,z−]→MN(A) completely given by z↦ u and ↦ 1N .

Denition .. (¿e algebraic Chern character)With the previous notations, the Chern character [ch(p)] ∈ HP(A) of the projector p is the imageof ∈ HP(C) in the composite map

HP(C)ip♯Ð→ HP(MN(A)) Tr♯Ð→ HP(A)

¿e Chern character [ch(u)] ∈ HP(A) of the invertible u is the image of ∈ HP(C[z,z−]) inthe composite map

HP(C[z,z−]) iu♯Ð→ HP(MN(A)) Tr♯Ð→ HP(A) ⧫

Proposition .. (¿e Chern character on algebraic K-theory)¿e class [ch(p)] ∈ HP(A) (resp. [ch(u)] ∈ HP(A)) depends only on the class of p in Kalg

(A)(resp. on the class of u in Kalg

(A)).¿e Chern character is a map ch ∶ Kalg

ν (A)→ HPν(A) for ν = , .

Example .. (Explicit formula for ch(p) inΩ(A))In order to give an explicit formula for the representative ch(p) of the class of the Chern character inthe mixed complex (Ω(A),bH,B), one has to explicitly write down the generator ∈ HP(C) = C.It is convenient to do that in the same mixed bicomplex (Ω(C),bH,B). In order to make notationsclear, let us denote by e ∈ C the unit element. ¿en one can show, using explicit formulas on bH andB, that

e +∑n≥

(−)n (n)!n!

(e − ) (de)n

is the generator of the class , in the total complex of the mixed complex (Ω(C),bH,B).Using the composite map at the level of mixed bicomplexes (ip and Tr), one gets

ch(p) = Tr(p) +∑n≥

(−)n (n)!n!

Tr((p− ) (dp)n) ⧫

Example .. (Explicit formula for ch(u) inΩU(A))

In the mixed bicomplex (ΩU(A),bH,B), we can give an explicit formula for the representative

ch(u) using the following expression of the representative of ∈ HP(C[z,z−]) = C:

∑n≥n!z−dz(dz−dz)n

¿en the element ch(u) takes the form

ch(u) =∑n≥n! Tr (u−du(du−du)n) ⧫


Remark .. (What is really a representative of the Chern character?)¿e Chern character is well dened only in the periodic cyclic homology of the algebra. But it isconvenient to manipulate it as a cycle in the complex computing this homology.

But which complex to consider? Indeed, as we saw before, there are many possibilities, at leastas many mixed bicomplexes that are b-quasi-isomorphic (Proposition ..). So that one can ex-pect some representative cycles in the complexesCCper

,(A), (Ω(A),b,B), (ΩU(A),b,B), and even

(ΩA∣C,,dK) if the algebra is a smooth commutative algebra. . .¿e representatives given in Examples .. and .. are then only particular expressions. For

instance, for the algebra A = SV, one can use the Kähler dierential calculus, in which any elementof degree ≥ dimV is . In that case, the Chern character is represented by a nite sum of dierentialforms of odd or even degrees.

¿e expressions we gave above have the advantage that they are written in the universal dier-ential calculi, in which all the degrees can be represented. Let us give another expression for thegenerator ∈ HP(C[z,z−]) in the bicomplex CCper

,(C[z,z−]). In order to do that, dene the fam-ily of elements

αn = (n + )!(z− − )⊗ (z − )⊗ [(z− − )⊗ (z − )]⊗n ∈ C[z,z−]⊗n+

βn = (n + )!(z − )⊗ [(z− − )⊗ (z − )]⊗n ∈ C[z,z−]⊗n+

then the representative of the generator is

c =∑n≥αn ⊕ βn ∈ TCCper

(C[z,z−])

Using the identication Ωn+(C[z,z−]) = C[z,z−]⊗n+ ⊕C[z,z−]⊗n+, this generator is also di-rectly written as a generator in the mixed bicomplex (Ω(C[z,z−]),b,B).

Finally, notice that the explicit development of the Chern character in one of the complexesmen-tioned above is completely determined by the lowest degrees, in which a normalisation is imposed,and the condition to be a cycle in the periodic complex. Hence this object is a very canonical one.⧫

Proposition .. (Naturality of the Chern character)For any short exact sequence of associative algebras 0 //I //A //A/I //0 , one has the com-mutative diagram

Kalg (I)

ch

// Kalg (A)

ch

// Kalg (A/I)

ch

δ // Kalg (I)

ch

// Kalg (A)

ch

// Kalg (A/I)

ch

HP(I) // HP(A) // HP(A/I) δ // HP(I) // HP(A) // HP(A/I)

(..)

Remark .. (¿e topological case)When the algebra is a topological algebra, one can show that the Chern character is in fact a mapfrom the K-groups dened on topological algebras and the continuous cyclic periodic homology.Indeed, one can show that it is homotopic invariant.

Nevertheless, remember that in Remark .. we mentioned that K-theory is well adapted toC∗-algebras and continuous functional calculus in general, but that cyclic periodic homology is onlyuseful for topological algebras underlying some smooth structures. . .

If one wants to connect K-theory and cyclic periodic homology directly at the level of repre-sentative cycles, one has to consider some intermediate algebras between “algebraic” and “C∗”, for


instance Fréchet or locally convex algebras. In these cases, unfortunately, the K-groups are not de-ned using projections and unitaries, so that the interpretation of the Chern character is not at alltransparent whereas it looks so clear in the algebraic version. . .

When the Bott periodicity takes place in K-theory, the commutative diagram (..) connects inreality the two six term exact sequences of Propositions .. and ... But there is a defect in thisclosed relation, a factor πi is necessary in the morphism δ ∶ K(A/I)→ K(I) to get a commutativediagram (see [Cuntz et al., ]).

Remark .. (¿e Chern character as an isomorphism)In ¿eorem .., we saw that the Chern character realizes an isomorphism between K-theory oftopological spaces (in fact its torsion-free part) and the ordinary cohomology of the underlyingtopological space.

In [Solleveld, ], it is shown that the Chern character for topological algebras realizes anequivalent isomorphism for a large class of Fréchet algebras in the following form

ch⊗Id ∶ K(A)⊗C ≃Ð→ HP(A)

In particular, the Fréchet algebras C∞(M) for a locally compact manifoldM is in this class. ⧫

Remark .. (Chern character and cyclic cohomology)For a Banach algebra A, the Chern character can be realized as a pairing Kν(A) × HPν(A) → C,using the natural pairing between periodic cyclic cohomology and periodic cyclic homology.

Let us consider the case ν = . In Example .., we dened cyclic cohomology using theConnescomplex. Let ϕ ∈ Cn

λ (A) be a cyclic cocycle and p ∈MN(A) a projector. Dene

⟨[p], [ϕ]⟩ = (iπ)nn! ∑i,...,iN

ϕ(pii , pii , . . . , piN i)

One can show that this pairing is well dened at the level of theK group and at the level ofHCn(A),and that it satises ⟨[p],S[ϕ]⟩ = ⟨[p], [ϕ]⟩. Because the periodic cyclic cohomology groupHP(A)can be dened as an inductive limit through the periodic operator S on the HCn(A) spaces, theprevious pairing is indeed a pairing between K(A) and HP(A).

Using this construction, no extra structure is required. One then recovers that the Chern charac-ter is indeed a canonical object in the context of K-theory and periodic cyclic (co)homology. ¿ereis a similar expression for ν = . ⧫

Remark .. (Comparing Chern characters)¿e expressions in Examples .., .., .. and .. look very similar. But there are dierenceswhich are important to be noted. In order to make them clear, we will call “geometric Chern charac-ter” the expressions given in Examples .., .. (and also ..), and “algebraic Chern character”the expressions given in Examples .. and ...

First, the spaces on which these Chern characters are expressed as “dierential forms” are notthe same in the two situations. In the geometric one, it is the de Rham dierential calculus. In thealgebraic one, it is one of the universal dierential calculi.

In order to compare them, one has to take into account a situation inwhich they bothmake sense,the case of the algebra A = C∞(M) for instance. In that case, one knows that the identicationof the Hochschild homology with de Rham forms can be expressed as in Example ... Using


these expressions, one easily show that the following squares are commutative, where the verticalisomorphisms concerning K-theories express the Serre-Swan¿eorem ..,

K(C∞(M))≃

chalg // HPcont (C∞(M))ϕ≃

K(M) chgeom // Heven(M;Q)

K(C∞(M))≃

chalg // HPcont (C∞(M))ϕ≃

K−(M) chgeom // Hodd(M;Q)¿is explains the extra factors used in the isomorphism ϕ in Example ... Notice that the two def-initions of the Chern characters are constrained: the geometric case is normalised in such a way thatit is a ring morphism, the algebraic one is expressed as an innite cycle in cyclic periodic homology,so that all the terms are normalised by the rst one. ¿e only degree of freedom in this square is theisomorphism of vector spaces ϕ (and fortunately not an isomorphism of algebras since HPν(A) hasno natural structure of algebra). ⧫

1.6 Conclusion

¿ere cannot be any conclusion to a subject that is still full of vivacity! ¿ousands of mathematicianstry everyday to conquest some new landmarks in this extraordinary vast and rich world. In thislecture, only some selected aspects of this theory have been presented. For instance, no mention hasbeen made about “noncommutative measure theory”, in which von Neumann algebras play the roleof C∗-algebras for measurable spaces.

We have seen that one can reasonably manipulate “noncommutative topological spaces” usingthe K-theory of C∗-algebras. One can convince oneself that dierentiable structures are available inthe heart of cyclic homology.

Nevertheless this research project is facing a challenge which have not yet been solved: what isthe noncommutative counterpart of smooth functions? Does it exist? We have made it clear thatcyclic homology sees some smooth structures, but the right category of “noncommutative smoothalgebras” has not yet been identied. Some paths have been investigated. For instance, Cuntz hasconsideredm-algebras (see [Fragoulopoulou, ]), some kind of locally convex algebras, onwhichhe succeed to enrichK-theory and cyclic homology (see [Cuntz et al., ]). Butwhat is stillmissingis a Gelfand-Neumark theorem for smooth functions.

2

SU(n)-principal fiber bundles andnoncommutative geometry

2.1 Introduction

¿e geometry of ber bundles is now widely used in the physical literature, mainly through theconcept of connections, which are interpreted as gauge elds in particle physics. It is worth to recallwhy the structure of these gauge theories leads to this mathematical identication. ¿e main pointswhich connect these two concepts are the common expression for gauge transformations and theeld strength of the gauge elds recognized as the curvature of the connection.

Since the introduction of the Higgs mechanics, some attempts have been made to understand itsgeometrical origin in a same satisfactory and elegant way as the gauge elds. ¿e reduction of somehigher dimensional gauge eld theories to somemore “conventional” dimensions has been proposedto reproduce the Higgs part of some models.

Nevertheless, one of the more convincing constructions from which Higgs elds emerged nat-urally and without the need to perform some dimensional reduction of some extra ad-hoc a er-ward arbitrary distortion of the model, was rstly exposed in [Dubois-Violette et al., a], andhighly popularized in subsequent work by A. Connes in its noncommutative standard model (see[Chamseddine and Connes, ] for a review of the recent developments in this direction). Whatthe pioneer work by Dubois-Violette, Kerner and Madore revealed is that the Higgs elds can beidentied with the purely noncommutative part of a noncommutative connection on an noncom-mutative algebra “containing” an ordinary smooth algebra of functions over a manifold and a purelynoncommutative algebra.

¿e algebra used there is the tensor product C∞(M)⊗Mn of smooth functions on some man-ifoldM and the matrix algebra of size n. ¿is “trivial” product does not reveal the richness of thisapproachwhen somemore intricate algebra is involved. In this review, we consider the algebra of en-domorphisms of a SU(n)-vector bundle. ¿is algebra reduces to the previous situation for a trivialvector bundle. ¿is non triviality gives rise to some elegant and powerful constructions we exposedin a series of previous papers, and to some results nowhere published before.

¿e rst part deals with some reviews of the ordinary geometry of ber bundles and connections.We think this is useful to x notations, but also to highlight what the noncommutative dierentialgeometry dened in the following extends from these constructions.

We then dene the general settings of our noncommutative geometry, which is based on deriva-

66 Chapter 2 – SU(n)-principal fiber bundles and noncommutative geometry

tions. ¿e notion of noncommutative connections is exposed, and some important examples arethen given to better understand the general situation.

¿e algebra we are interested in is then introduced as the algebra of endomorphisms of a SU(n)-vector bundle. We show how it is related to ordinary geometry, and how ordinary connections playsan essential role to study its noncommutative geometry.

¿e noncommutative connections on this algebra are then studied, and here we recall why thepurely noncommutative part can be identied with Higgs elds.

¿en it is shown that this algebra is indeed related, through the algebraic notion of Cartan op-erations on a bigger algebra, to the geometry of the SU(n)-principal ber bundle underlying thegeometry of the SU(n)-vector bundle.

Some considerations about the cohomology behind the endomorphism algebra are then ex-posed, in particular a new construction of the Chern classes of the SU(n)-vector bundle whichare obtained from a short exact sequence of Lie algebras of derivations.

¿e last section is concerned with the symmetric reduction of noncommutative connections,which generalizes a lot of previous works about symmetric reduction of ordinary connections.

2.2 A brief review of ordinary fiber bundle theory

¿e noncommutative geometry we will consider in the following contains, and relies in an essentialway to the ordinary dierential geometry of the SU(n)-ber bundles theory. ¿is section is devotedto some aspects of this dierential geometry. Its aim is to x notations but also to present someconstructions which will be generalized or completed by the noncommutative geometry introducedlater on.

2.2.1 Principal and associated fiber bundles

LetM be a smoothmanifold andG a Lie group. Denote by G //P π //M a (locally trivial) principalber bundle for the right action of G on P , denoted by p↦ p⋅g = Rgp.

For any p ∈ P , one denes Vp = Ker(Tpπ ∶ TpP → Tπ(p)M), the vertical subspace of TpP . Forany X ∈ g, let

Xv(p) = (ddtp⋅ exp(tX))

t=

¿en Vp = Xv(p)/X ∈ g and one has Rg∗Vp = Vp⋅g.¿is denes vertical vector elds over P and horizontal dierential forms, which are dierential

forms on P which vanish when one of its arguments is vertical.Let (U,ϕ) be a local trivialisation of P over an open subset U ⊂ M, which means that there

exists a isomorphism ϕ ∶ U × G ≃Ð→ π−(U) such that π(ϕ(x,h)) = x and ϕ(x,hg) = ϕ(x,h)⋅g forany x ∈ U and g,h ∈ G.

If (Ui,ϕi) and (Uj,ϕj) are two local trivialisations such that Ui ∩ Uj ≠ ∅, then there exists adierentiable map gi j ∶ Ui ∩ Uj → G such that, if ϕi(x,hi) = ϕj(x,hj) for hi,hj ∈ G, then hi =gi j(x)hj for any x ∈ Ui ∩ Uj. ¿e gi j are called the transition functions for the system (Ui,ϕi)iof local trivialisations. ¿ey satisfy gi j(x) = g−ji (x) for any x ∈ Ui ∩ Uj and the cocycle conditiongi j(x)gjk(x)gki(x) = e for any x ∈ Ui ∩Uj∩Uk ≠ ∅.

Now, let F be a manifold on which G acts on the le : φ ↦ ℓgφ. On the manifold P × F wedene the right action (p,φ) ↦ (p⋅g, ℓg−φ), and we denote by E = (P × F)/G the orbit space for

2.2 A brief review of ordinary fiber bundle theory 67

this action. ¿is is the associated ber bundle toP for the couple (F , ℓ). It is denoted by E = P ×ℓF ,and [p,φ] ∈ E is the projection of (p,φ) in the quotient P ×F → (P ×F)/G. By construction, onehas [p⋅g,φ] = [p, ℓgφ].

A (smooth) section of E is a (dierentiable) map s ∶M→ E such that πs(x) = x for any x ∈M.We denote by Γ(E) the space of dierentiable sections of E .

¿e main point of this construction is the fact that one can show that Γ(E) identies with thespace FG(P ,F) = Φ ∶ P → F / Φ(p⋅g) = ℓg−Φ(p) of G-equivariant maps P → F . ¿is resultwill be useful later on.

Let (U,ϕ) be a local trivialisation of P overU. ¿en the smooth map sU ∶ U → π−(U) given bysU(x) = ϕ(x, e) is a local section of P .

Any section s of P is locally given by a local map h ∶ U → G such that s(x) = sU(x)⋅h(x) =ϕ(x,h(x)). We call sU a local gauge over U for P , because it is used as a local reference in P todecompose sections on P .

In the same way, any section s of E is locally given by a local map φ ∶ U → F such that s(x) =[sU(x),φ(x)]. ¿is means that the local gauge sU can also be used to decompose sections of anyassociated ber bundle.

Let (Ui,ϕi) and (Uj,ϕj) be two local trivialisations such that Ui ∩Uj ≠ ∅. ¿en one has

sj(x) = ϕj(x, e) = ϕi(x, gi j(x)) = ϕi(x, e)⋅gi j(x) = si(x)⋅gi j(x)

so that on P , if s(x) = si(x)⋅hi(x) = sj(x)⋅hj(x), then

hi(x) = gi j(x)hj(x)

On E , if s(x) = [si(x),φi(x)] = [sj(x),φj(x)] for x ∈ Ui ∩Uj ≠ ∅, then

φi(x) = ℓgij(x)φj(x)

¿ese are the transformation laws for the local decompositions of sections in P and E .Let F be a vector space and ℓ a representation (linear action) of G. In that case, the associated

ber bundle E for the couple (F, ℓ) is called a vector bundle. ¿e space of smooth sections Γ(E) isthen a C∞(M)-module for the pointwise multiplication: f(x)s(x) for any f ∈ C∞(M), s ∈ Γ(E)and x ∈M.

Moreover, if E and E ′ are vector bundles, then E∗ (dual), E ⊕ E ′ (Whitney sum), E ⊗ E ′ (tensorproduct) and ⋀E (exterior product) are dened. ¿ey are associated respectively to (F∗, ℓ∗), (F ⊕F′, ℓ⊕ ℓ′), (F ⊗ F′, ℓ⊗ ℓ′) and (⋀F,⋀ℓ).

Here are now the main examples which will be at the root of the noncommutative geometry in-troduced in the following, and will permits one to make connections between this noncommutativegeometry and the pure geometrical context.

Example .. (Tangent and cotangent spaces)¿e tangent space TM → M, and the cotangent space T∗M → M are canonical vector bundlesoverM.

¿e space Γ(TM) will be denoted by Γ(M). It is the C∞(M)-module of vector elds onM. Itis also a Lie algebra for the bracket [X,Y]⋅f = X⋅Y⋅f − Y⋅X⋅f for any f ∈ C∞(M).

One the other hand, by duality, Γ(T∗M) = Ω(M) is the space of -forms onM. Extendingthis construction, Γ(⋀T∗M) = Ω(M) is the algebra of (de Rham) dierential forms onM.


If E is a vector bundle overM, then Γ(T∗M⊗ E) = Ω(M,E) is the space of dierential formswith values in the vector bundle E , whichmeans that for any ω ∈ Ωp(M,E), Xi ∈ Γ(M) and x ∈M,ω(X, . . . ,Xp)(x) ∈ Ex. ⧫

Example .. (¿e endomorphism bundle)Consider the case where F is a nite dimensional vector space. ¿en E∗ ⊗ E is associated to P forthe couple (F∗ ⊗ F, ℓ∗ ⊗ ℓ).

One has the identication F∗⊗F ≃ End(F) by (α⊗φ)(φ′) = α(φ′)φ, where End(F) is the spaceof endomorphisms of F.

¿e vector bundle End(E) = E∗ ⊗ E is called the endomorphism ber bundle of E .¿ere is a natural pairing Γ(E∗)⊗Γ(E)→ C∞(M) denoted by x ↦ ⟨α(x), s(x)⟩. One can show

that Γ(E∗ ⊗ E) = Γ(End(E)) is an algebra, which identies with Γ(E∗) ⊗C∞(M) Γ(E) and with thespace of C∞(M)-module maps Γ(E)→ Γ(E) by (α⊗ s)(s′)(x) = ⟨α(x), s′(x)⟩s(x). ⧫

Example .. (¿e gauge group and its Lie algebra)¿e group G acts on itself by conjugaison: αg(h) = ghg−. ¿e associated ber bundle P ×α G hasG as ber but is not a principal ber bundle. In particular, this ber bundle has a global section,dened in any trivialisation by x ↦ e, where e ∈ G is the unit element. But one knows that theexistence of a global section on P is equivalent to P being trivial.

Denote by G = Γ(P ×α G) the space of smooth sections. It is a group, called the gauge groupof P : it is the sub-group of vertical automorphisms in Aut(P), the group of all automorphisms ofP . Indeed, any element in G is also a G-equivariant map Φ ∶ P → G, which denes the verticalautomorphism p↦ p⋅Φ(p). ¿e compatibility condition is ensured by the G-equivariance: p⋅g ↦(p⋅g)⋅Φ(p⋅g) = (p⋅g)⋅(g−Φ(p)g) = (p⋅Φ(p))⋅g.

By construction, one has the short exact sequence of groups:

1 //G //Aut(P) //Aut(M) //1

G acts on the vector space g by the adjoint action Ad. Denote by AdP = P ×Ad g the associatedvector bundle. ¿e vector space Γ(AdP) is the Lie algebra of the gauge group G, denoted herea erby LieG. ⧫

2.2.2 Connections

Let G //P π //M be a principal ber bundle, and let E →M be an associated vector bundle. ¿ereis at least three ways to dene a connection in this context:

Geometrical denition: A connection on P is a smooth distributionH in TP such that for anyp ∈ P and g ∈ G:

TpP = Vp⊕Hp and Rg∗Hp = Hp⋅g

¿is denes horizontal vector elds and vertical dierential forms (forms which vanish whenone of its arguments is horizontal).

One gets the geometrical notion of horizontal li ing of vector elds onM, which we denoteby Γ(M) ∋ X↦ Xh ∈ Γ(P).

2.2 A brief review of ordinary fiber bundle theory 69

Algebraic denition: A connection on P is a -form on P taking values in the Lie algebra g,ω ∈ Ω(P)⊗ g, such that for any g ∈ G and X ∈ g:

R∗gω = Adg−ω (equivariance) and ω(Xv) = X (vertical condition)

¿e associated horizontal distribution is Hp = Kerω∣p.

Analytic denition: A connection on E is a linear map ∇EX ∶ Γ(E) → Γ(E) dened for anyX ∈ Γ(M), such that for any f ∈ C∞(M), s ∈ Γ(E), X,Y ∈ Γ(M):

∇EX(fs) = (X⋅f)s + f∇EXs ∇EfXs = f∇EXs ∇EX+Ys = ∇EXs +∇EYs

If s ∈ Γ(E) corresponds toΦ ∈ FG(P ,F), then ∇EXs corresponds to Xh⋅Φ.

¿e equivariance of the connection -form ω implies the relation

LXvω+ [X,ω] =

for any X ∈ g.For each of these three denitions, the curvature of a connection can be introduced:

Geometrical denition: ¿ere exists a geometrical interpretation of the curvature as the ob-struction to the closure of horizontal li s of “innitesimal” closed paths onM.

Let γ ∶ [, ] ↦M be a closed path and let p ∈ P . ¿ere exists a unique path γh ∶ [, ] ↦ Psuch that γh() = pand γh(t) ∈ Hγh(t) for any t ∈ [, ]. γh is a horizontal li ing of γ. One hasγh() ≠ γh() = pa priori, but they are in the same ber, so that the deciency is in G.When the path γ is shrunk to an innitesimal path, the deciency is an element in g whichdepends only on γ() and γ(). ¿is is the curvature.

Algebraic denition: ¿e curvature is the equivariant horizontal -formΩ ∈ Ω(P)⊗g denedfor any X ,Y ∈ Γ(P) by

Ω(X ,Y) = dω(X ,Y) + [ω(X ),ω(Y)]

It satises the Bianchi identitydΩ+ [ω,Ω] =

Analytic denition: Given ∇EX ∶ Γ(E) → Γ(E), the curvature RE(X,Y) is the map dened forany X,Y ∈ Γ(M) by

RE(X,Y) = ∇EX∇EY −∇EY∇EX −∇E[X,Y] ∶ Γ(E)→ Γ(E)

¿e remarkable fact is that this particular combinaison is a C∞(M)-module map.

One can connect these denitions by the following relations. Let η be the representation of g onF induced by the representation ℓ of G. If s ∈ Γ(E) corresponds to Φ ∈ FG(P ,F), then RE(X,Y)scorresponds to η(Ω(X ,Y))⋅Φ for any X ,Y such that π∗X = X and π∗Y = Y.


Globally onP Locally onMω ∈ Ω(P)⊗g, equivariant, verticalcondition.

Family of local -forms Aii, Ai ∈ Ω(Ui)⊗g, satisfy-ing gluing non homogeneous relations.

Ω ∈ Ω(P) ⊗ g, equivariant andhorizontal.

Family of local -forms Fii, Fi ∈ Ω(Ui)⊗ g, satisfy-ing gluing homogeneous relations.

Table .: ¿e two ordinary constructions of the connections and curvature, the global one onP andthe local one onM.

Let ω ∈ Ω(P) ⊗ g be a connection -form on P , and Ω its curvature. Let (U,ϕ) be local trivi-alisation of P , and s its associated local section.

One can dene the local expression of the connection and the curvature in this trivialisation asthe pull-back of ω and Ω by s ∶ U → P :

A= s∗ω ∈ Ω(U)⊗ g F = s∗Ω ∈ Ω(U)⊗ g

If (Ui,ϕi) and (Uj,ϕj) are two local trivialisations, onUi∩Uj ≠ ∅ one has the well-known relations

Aj = g−i jAigi j+ g−i jdgi j Fj = g−i jFigi j (..)

with obvious notations. A family of -forms Aii satisfying these gluing relations denes a con-nection -form on P . ¿is is (too) o en used in the physical literature as a possible denition of aconnection and its curvature.Remark .. (Intermediate construction)We summarize in Table . the two common ways to introduce a connection as dierential objects,either as a global -form on P or as a family of local -forms onM.

It is well known that, using the homogeneous gluing relations for the Fi’s, or using the equivariantand horizontal property of Ω, one can show that the curvature is also a section of the associatedvector bundle ⋀T∗M ⊗AdP , i.e. a global -forms onM with values in the vector bundle AdP =P ×Ad g. We denote by F ∈ Ω(M,AdP) this -form.

Because of the inhomogeneous gluing relations for theAi’s, the connection cannot be the sectionof such an “intermediate” construction between forms on P and local forms on the Ui’s.

Let us mention here that in the noncommutative geometry introduced in the following, thisintermediate construction is possible also for the connection -form. See Remark ... ⧫

2.2.3 Gauge transformationsWe saw that the gauge groupG = Γ(P×αG) acts onP . To any a ∈ G one can associate aG-equivariantmap Φ ∶ P → G. ¿e corresponding vertical dieomorphism P → P dened by a is also denotedby a

Let ω ∈ Ω(P)⊗g be a connection onP . ¿en one can show that the pull-back a∗ω is also a con-nection and a∗Ω is its curvature. Explicitly, one can establish the formulae a∗ω = Φ−ωΦ+Φ−dΦand a∗Ω = Φ−ΩΦ, which look very similar to (..), but are not the same: here we perform someactive transformation on the space of connections while in (..) we look at the same connection indierent trivialisations. ¿is is the dierence between active and passive transformation laws.

In order to get the action of the Lie algebra of the gauge group, considerΦ = exp(ξ)with ξ ∶ P →g, G-equivariant, so that ξ denes an element in LieG = Γ(AdP). ¿en the innitesimal action onconnections and curvatures take the form:

ω↦ dξ + [ω, ξ] Ω↦ [Ω, ξ]

2.3 Derivation-based noncommutative geometry 71

2.3 Derivation-based noncommutative geometry

In this section, we introduce the algebraic context inwhich the noncommutative geometry we are in-terested in is constructed. ¿e dierential calculus we consider here has been introduced in [Dubois-Violette, ] and has been exposed and studied for various algebras, for instance in [Dubois-Violette et al., b], [Dubois-Violette et al., a], [Masson, ], [Masson, ], [Dubois-Violette and Masson, ],[Masson, ], [Dubois-Violette and Michor, ], [Dubois-Violetteand Michor, ], [Dubois-Violette and Michor, ].

2.3.1 Derivation-based differential calculusLet A be an associative algebra with unit 1. Denote by Z(A) the center of A.

Denition .. (Vector space of derivations of A)¿e vector space of derivations ofA is the space Der(A) = X ∶ A→ A /X linear,X(ab) = X(a)b+aX(b),∀a,b ∈ A ⧫

¿e essential properties of this space are contained in the following:

Proposition .. (Structure of Der(A))Der(A) is a Lie algebra for the bracket [X,Y]a = XYa−YXa (∀X,Y ∈ Der(A)) and aZ(A)-modulefor the product (fX)a = f(Xa) (∀f ∈ Z(A), ∀X ∈ Der(A)).

¿e subspace Int(A) = ada ∶ b ↦ [a,b] / a ∈ A ⊂ Der(A), called the vector space of innerderivations, is a Lie ideal and a Z(A)-submodule.

WithOut(A) = Der(A)/Int(A), there is a short exact sequence of Lie algebras andZ(A)-modules

0 //Int(A) //Der(A) //Out(A) //0 (..)

In case A has an involution a↦ a∗, one can dene real derivations:

Denition .. (Real derivations for involutive algebras)If A is an involutive algebra, the derivation X ∈ Der(A) is real if (Xa)∗ = Xa∗ for any a ∈ A. Wedenote by DerR(A) the space of real derivations. ⧫

Denition .. (¿e graded dierential algebraΩDer(A))

LetΩnDer(A) be the set ofZ(A)-multilinear antisymmetricmaps fromDer(A)n toA, withΩ

Der(A) =A, and let

ΩDer(A) =⊕n≥Ωn

Der(A)

We introduce on ΩDer(A) a structure of N-graded dierential algebra using the product

(ωη)(X, . . . ,Xp+q) =p!q! ∑

σ∈Sp+q

(−)sign(σ)ω(Xσ(), . . . ,Xσ(p))η(Xσ(p+), . . . ,Xσ(p+q))

and using the dierential d (of degree ) dened by the Koszul formula

dω(X, . . . ,Xn+) =n+∑i=

(−)i+Xiω(X, ⋅ ⋅ ⋅i∨. . . . ,Xn+)

+ ∑≤i<j≤n+

(−)i+jω([Xi,Xj], ⋅ ⋅ ⋅i∨. ⋅ ⋅ ⋅

j∨. . . . ,Xn+) ⧫


Denition .. (¿e graded dierential algebraΩDer(A))

Denote by ΩDer(A) ⊂ Ω

Der(A) the sub dierential graded algebra generated in degree by A. ⧫

Notice that by denition, every element inΩnDer(A) is a sum of terms of the form ada⋯dan for

a, . . . ,an ∈ A.

¿e previous denitions are motivated by the following important example which shows thatthese denitions are correct generalisations of the space of ordinary dierential forms on amanifold:

Example .. (¿e algebra A = C∞(M))LetM be a smooth manifold and let A = C∞(M). ¿e center of this algebra is A itself: Z(A) =C∞(M). ¿e Lie algebra of derivations is exactly the Lie algebra of smooth vector elds onM:Der(A) = Γ(M). In that case, there is no inner derivations, Int(A) = 0, so that Out(A) = Γ(M).

¿e two graded dierential algebras coincide with the graded dierential algebra of de Rhamforms onM: Ω

Der(A) = ΩDer(A) = Ω(M) ⧫

In the previous denitions of the graded dierential calculi, one is not bounded to consider theentire Lie algebra of derivations:

Denition .. (Restricted derivation-based dierential calculus)Let g ⊂ Der(A) be a sub Lie algebra and a subZ(A)-module. ¿e restricted derivation-based dier-ential calculus Ω

g(A) associated to g is dened as the set of Z(A)-multilinear antisymmetric mapsfrom gn to A for n ≥ , using the previous formulae for the product and the dierential. ⧫

Now, let g be any Lie subalgebra of Der(A). ¿en g denes a natural operation in the senseof H. Cartan on the graded dierential algebra (Ω

Der(A),d). ¿e interior product is the gradedderivation of degree − on Ω

Der(A) dened by

iX ∶ ΩnDer(A)→ Ωn−

Der(A) (iXω)(X, . . . ,Xn−) = ω(X,X, . . . ,Xn−)

∀X ∈ g, ∀ω ∈ ΩnDer(A) and ∀Xi ∈ Der(A). By denition, iX is on Ω

Der(A) = A.¿e associated Lie derivative is the graded derivation of degree on Ω

Der(A) given by

LX = iXd + diX ∶ ΩnDer(A)→ Ωn

Der(A)

One can easily verify the relations dening a Cartan operation:

iXiY + iYiX = LXiY − iYLX = i[X,Y]

LXLY − LYLX = L[X,Y] LXd − dLX =

One can then associate to this operation the following subspaces of ΩDer(A):

• ¿e horizontal subspace is the kernel of all the iX for X ∈ g. ¿is is a graded algebra.

• ¿e invariant subspace is the kernel of all the LX forX ∈ g. ¿is is a graded dierential algebra.

• ¿e basic subspace is the kernel of all the iX and LX for X ∈ g. ¿is is a graded dierentialalgebra.

For instance, g = Int(A) denes such an operation.


2.3.2 Noncommutative connections and their propertiesNoncommutative connections play a central role in noncommutative dierential geometry. ¿eyare all based on some generalisation of what we called the analytic denition of ordinary connec-tions, where one replaces the C∞-module of sections of a vector bundle by a more general (nitelyprojective) module over the algebra. Various denitions has been proposed, for instance to take intoaccount some bimodule structures. Here we only consider right A-modules.

Definitions and general properties

LetM be a right A-module.

Denition .. (Noncommutative connections, curvature)A noncommutative connection on M for the dierential calculus based on derivations is a linearmap ∇X ∶ M → M, dened for any X ∈ Der(A), such that ∀X,Y ∈ Der(A), ∀a ∈ A, ∀m ∈ M,∀f ∈ Z(A):

∇X(ma) = m(Xa) + (∇Xm)a, ∇fXm = f∇Xm, ∇X+Ym = ∇Xm + ∇Ym

¿e curvature of ∇ is the linear map R(X,Y) ∶ M→ M dened for any X,Y ∈ Der(A) by

R(X,Y)m = [∇X, ∇Y]m − ∇[X,Y]m ⧫

¿is denition is an adaptation to the derivation-based noncommutative calculus of the ordinary(analytic) denition of connections. Notice that we have to make use of the center Z(A) of thealgebra A for one of the above relations, which means that we have to dierentiate the respectiveroles of the algebra and of its center.

Proposition .. (General properties)¿e space of connections is an ane space modeled over the vector space HomA(M,M ⊗A Ω

Der(A))(right A-module morphisms fromM toM⊗A Ω

Der(A)).R(X,Y) ∶ M→ M is a right A-module morphism.

Denition .. (¿e gauge group)¿e gauge group ofM is the group of automorphisms ofM as a right A-module. ⧫

Proposition .. (Gauge transformations)For anyΦ in the gauge group ofM and any noncommutative connection ∇, themap ∇Φ

X = Φ−∇XΦ ∶M→ M is a noncommutative connection.

¿is denes the action of the gauge group on the space of noncommutative connections.

Suppose now that A is an involutive algebra and let as beforeM be a right A-module.

Denition .. (Hermitean structure, compatible noncommutative connections)A Hermitean structure on M is a sesquilinear form ⟨−,−⟩ ∶ M ×M → A such that, ∀m,m ∈ M,∀a,a ∈ A,

⟨m,m⟩∗ = ⟨m,m⟩ ⟨ma,ma⟩ = a∗ ⟨m,m⟩a

A noncommutative connection ∇ is compatible with ⟨−,−⟩ if, ∀m,m ∈ M, ∀X ∈ DerR(A),

X⟨m,m⟩ = ⟨∇Xm,m⟩ + ⟨m, ∇Xm⟩ ⧫


Denition .. (“Unitary” gauge transformations)An elementΦ in the gauge group is compatible with the Hermitean structure if, for anym,m ∈ M,one has ⟨Φ(m),Φ(m)⟩ = ⟨m,m⟩. In that case, we refer to such a gauge transformation as a“unitary” gauge transformation. ⧫

Lemma ..¿e space of compatible noncommutative connections with ⟨−,−⟩ is stable under “unitary” gauge trans-formations.

The right A-module M = AAs a special case of the previous general situation, we consider the right A-module M = A. Let∇X ∶ A→ A be a noncommutative connection.

Proposition .. (Noncommutative connections onM = A)¿e noncommutative connection ∇ is completely determined by ∇X1 = ω(X), with ω ∈ Ω

Der(A), bythe relation

∇Xa = Xa + ω(X)a

¿e curvature of ∇ is the multiplication on the le on A by the noncommutative -form

Ω(X,Y) = dω(X,Y) + [ω(X),ω(Y)]

¿e gauge group is identied with the invertible elements g ∈ A by Φg(a) = ga and the gauge trans-formations on ∇ take the following form on ω and Ω:

ω↦ ωg = g−ωg+ g−dg Ω↦ Ωg = g−Ωg

∇X, dened by a↦ Xa, is a noncommutative connection on A.

¿e gauge transformations on the noncommutative forms ω andΩ are clearly of the same natureas the one encountered in ordinary dierential geometry. Nevertheless, the relations are dierent:the dierential operator is the noncommutative dierential here.

In the particular case when A is involutive, one can dene a canonical Hermitean structure onM by ⟨a,b⟩ = a∗b. ¿en, U(A) = u ∈ A / u∗u = uu∗ = 1, the group of unitary elements of A,identies with the unitary gauge group.

Let us stress the following important point.

Remark .. (Vector space versus gauge transformations)We saw that the space of noncommutative connections is an ane space, but here it looks like thevector spaceΩ

Der(A). In fact, one can show that gauge transformations are not compatible with thislinear structure:

(λω + λω)u = u−(λω + λω)u + u−duλωu + λωu = λ(u−ωu + u−du) + λ(u−ωu + u−du)

are not equal except for λ + λ = . ⧫

¿e following proposition applies in some important examples:


Proposition .. (Canonical gauge invariant noncommutative connection)If there exists a noncommutative -form ξ ∈ Ω

Der(A) such that da = [ξ,a] for any a ∈ A, then thecanonical noncommutative connection dened by ∇−ξ

X a = Xa − ξ(X)a can be written as ∇−ξX a =

−aξ(X).Moreover, this canonical noncommutative connection is gauge invariant.

Proof One has Xa = [ξ(X),a], so that ∇−ξX a = [ξ(X),a] − ξ(X)a = −aξ(X).

Let u ∈ U(A) be a unitary gauge transformation. Its action on the noncommutative -form −ξis (−ξ)u = −u−ξu+u−du = u−(−ξu+ [ξ,u]) = u−(−uξ) = −ξ, which shows that this noncommu-tative connection is indeed gauge invariant. ∎

As can be immediately seen, this situation can’t occur in the commutative case (ordinary dier-ential geometry) because for any -form ξ, one has [ξ,a] = . Below, we will encounter a situationwhere such a noncommutative -form exists, in the context of the algebra Mn(C) of complex ma-trices. An other important example where such an invariant noncommutative connection makes itsappearance is the Moyal algebra. ¿ese two examples share in common that they only have innerderivations. ¿ey are highly noncommutative situation in this respect, even if theMoyal algebra canbe considered as a deformation of some commutative algebra of ordinary smooth functions.

The right A-module M = AN

As an other special case of right A-modules, we consider now the case where the right A-moduleis M = AN . Denote by ei = (, . . . ,1, . . . ,), for i = , . . . ,N, a canonical basis of the right moduleAN . We look at m = eiai ∈ M as a column vector for the ai’s, so that we use some matrix productnotations. We also use the notation Xm = ei(Xai) for any derivation X of A.

Let ∇X ∶ AN → AN be a noncommutative connection.

Proposition .. (Noncommutative connections onM = AN)¿e noncommutative connection ∇ is completely determined by N noncommutative -forms ωj

i ∈ΩDer(A) dened by ∇Xei = ejωj

i(X), through the relation ∇Xm = Xm + ω(X)m, with ω = (ωji) ∈

MN(ΩDer(A)).

¿e curvature of ∇ is the multiplication on the le onAN by the matrix of noncommutative -formsΩ = dω+ [ω,ω] ∈MN(Ω

Der(A)).¿e gauge group of AN is GLN(A) (invertibles in MN(A)), which acts by le (matrix) multipli-

cation. ¿e gauge transformations take the forms ωg = g−ωg + g−dg and Ωg = g−Ωg in matrixnotations.

∇X, dened by m↦ Xm, is a noncommutative connection on AN.

In the particular case when A is involutive, the natural Hermitean structure onM is dened by⟨(ai),(bj)⟩ = ∑N

i=(ai)∗bi. ¿en, UN(A) = u ∈ MN(A) / u∗u = uu∗ = 1N, the group of unitaryelements ofMN(A), is the unitary gauge group.

The projective finitely generated right A-modules

From the Serre-Swan theorem, one knows that any vector bundle on a smooth manifoldM is char-acterised by its space of smooth sections as a projective nitely generated right module (p.f.g.m.)over C∞(M). ¿e natural generalisation of vector bundles in noncommutative geometry is thentaken to be the projective nitely generated right A-modules.

LetM be such a projective nitely generated rightA-modules. M is a direct summand inAN , sothat there exists a projection p ∈MN(A) such thatM = pAN .


Proposition .. (Noncommutative connections on p.f.g.m.)If ∇ is a noncommutative connection on the right A-module AN, then m↦ p∇Xm denes a noncom-mutative connection onM, where m ∈ M ⊂ AN.

¿e curvature of the noncommutative connection obtained this way from the canonical noncom-mutative connection ∇

X of Proposition .., is the multiplication on the le onM ⊂ AN by the matrixof noncommutative -forms pdpdp.

Example .. (¿e algebra A = C∞(M))We saw that the noncommutative derivation-based dierential calculus is the ordinary de Rhamcalculus. Using the equivalence given in the Serre-Swan theorem, the denitions of (ordinary) con-nections and of noncommutative connections coincide. ⧫

2.3.3 Two important examples

The algebra A = Mn(C) = Mn

Let us consider the caseA =Mn(C) =Mn, the nite dimensional algebra of n×n complex matrices.¿is is an involutive algebra for the adjointness of matrices.

First, we summarize the general properties of its derivation-based dierential calculus, which isdescribed in [Dubois-Violette, ], [Dubois-Violette et al., b] and [Masson, ].

Proposition .. (General properties of the dierential calculus)One has the following results:

• Z(Mn) = C.

• Der(Mn) = Int(Mn) ≃ sln = sl(n,C) (traceless matrices). ¿e explicit isomorphism associatesto any γ ∈ sln(C) the derivation adγ ∶ a↦ [γ,a].DerR(Mn) = su(n) and Out(Mn) = 0.

• ΩDer(Mn) = Ω

Der(Mn) ≃ Mn ⊗ ⋀sl∗n, with the dierential d′ coming from the dierential of

the dierential complex of the Lie algebra sln represented on Mn by the adjoint representation(commutator).

• ¿ere exits a canonical noncommutative -form iθ ∈ ΩDer(Mn) such that for any γ ∈Mn(C)

iθ(adγ) = γ − n Tr(γ)1

¿is noncommutative -form iθ makes the explicit isomorphism Int(Mn(C)) ≃Ð→ sln.

• iθ satises the relation d′(iθ)−(iθ) = . ¿is makes iθ look very much like theMaurer-Cartanform in the geometry of Lie groups (here SLn(C)).

• For any a ∈ Mn, one has d′a = [iθ,a] ∈ ΩDer(Mn). ¿is relation is no more true in higher

degrees.

Let us now introduce a particular basis of this algebra, which permits one to perform explicitcomputations. Denote by Ekk=,...,n− a basis for sln of hermitean matrices. ¿en, it denes abasis for the Lie algebra Der(Mn) ≃ sln through the n − derivations ∂k = adiEk , which are realderivations. Adjoining the unit 1 to the Ek’s, one gets a basis forMn.


Let us dene the θℓ’s in sl∗n by duality: θℓ(∂k) = δℓk. ¿en θℓ is a basis of -forms in ⋀sl∗n. Bydenition, they anticommute: θℓθk = −θkθℓ in this exterior algebra.

Dene the structure constants by [Ek,Eℓ] = −iCmkℓEm. ¿en one can show that the dierential d′

takes the explicit form:

d′1 = d′Ek = −CmkℓEmθ

ℓ d′θk = − Ckℓmθ

ℓθm

¿e noncommutative -form iθ can be written as iθ = iEkθk ∈ Mn ⊗ ⋀sl∗n. It is obviouslyindependent of the chosen basis.

Proposition .. (¿e cohomology of the dierential calculus)¿e cohomology of the dierential algebra (Ω

Der(Mn),d′) is

H(ΩDer(Mn),d′) = I(⋀sl∗n)

the algebra of invariant elements for the natural Lie derivative.Recall that the algebra I(⋀sl∗n) is the graded commutative algebra generated by elements cnr− in

degree r − for r ∈ ,, . . . ,n.

Let us introduce the symmetric matrix gkℓ = n Tr(EkEℓ). ¿en the gkℓ’s dene a natural metric

(scalar product) on Der(Mn) with the relation g(∂k,∂ℓ) = gkℓ.Now, one can show that every dierential form of maximal degree ω ∈ Ωn−

Der (Mn) can be writtenuniquely in the form

ω = a√

∣g∣θ⋯θn−

where a ∈Mn and where ∣g∣ is the determinant of the matrix (gkℓ).Denition .. (Noncommutative integration)One denes a noncommutative integration

∫n.c. ∶ ΩDer(Mn)→ C

by ∫n.c. ω = n Tr(a) if ω ∈ Ωn−

Der (Mn) written as above, and otherwise.¿is integration satises the closure relation

∫n.c. d′ω = ⧫

Let us now consider the right A-moduleM = A.¿enoncommutative -form−iθ denes a canonical noncommutative connection by the relation

∇−iθX a = Xa − iθ(X)a for any a ∈ A.

Proposition .. (Properties of ∇−iθ)For any a ∈Mn and X = adγ ∈ Der(Mn) (with Trγ = ), one has

∇−iθX a = −aiθ(X) = −aγ

∇−iθ is gauge invariant.¿e curvature of the noncommutative connection ∇−iθ is zero.


Proof ¿is is a consequence of the existence of the canonical gauge invariant noncommutativeconnection implied by the relation d′a = [iθ,a] (Proposition ..).

¿e curvature is the noncommutative -formΩ(X,Y) = d′(−iθ)(X,Y)+[(−iθ)(X),(−iθ)(Y)] =−(d′iθ(X,Y) − (iθ)(X,Y)) = . ∎

Let us now consider the right A-module M = Mr,n, the vector space of r × n complex matriceswith the obvious right module structure and the Hermitean structure ⟨m,m⟩ = m∗

m ∈Mn.

Proposition .. (∇−iθ, at noncommutative connections)¿e noncommutative connection ∇−iθ

X m = −miθ(X) is well dened, it is compatible with the Her-mitean structure and its curvature is zero.

Any noncommutative connection can bewritten ∇Xa = ∇−iθX a+A(X)a forA= Akθk withAk ∈Mr.

¿e curvature of ∇ is the multiplication on the le by theMr-valued noncommutative -form

F = ([Ak,Aℓ] − Cm

kℓAm)θkθℓ

¿is curvature vanishes if and only if A ∶ sln →Mr is a representation of the Lie algebra sln.Two at connections are in the same gauge orbit if and only if the corresponding Lie algebra repre-

sentations are equivalent.

For the proof, we refer to [Dubois-Violette et al., b].

The algebra A = C∞(M)⊗Mn

As a second important example, we consider now themixed of the two algebrasC∞(M) andMn(C)studied before, in the form of matrix valued functions on a smooth manifoldM (dimM = m).

¿e derivation-based dierential calculus for this tensor product algebra was rst considered in[Dubois-Violette et al., a]:Proposition .. (General properties of the dierential calculus)One has the following results:

• Z(A) = C∞(M).

• Der(A) = [Der(C∞(M)) ⊗ 1] ⊕ [C∞(M) ⊗ Der(Mn)] = Γ(M) ⊕ [C∞(M) ⊗ sln] as Liealgebras and C∞(M)-modules. In the following, we will use the notations: X = X + adγ withX ∈ Γ(M) and γ ∈ C∞(M)⊗ sln = A (traceless elements in A).One can identify Int(A) = A and Out(A) = Γ(M).

• ΩDer(A) = Ω

Der(A) = Ω(M) ⊗ ΩDer(Mn) with the dierential d = d + d′, where d is the

de Rham dierential and d′ is the dierential introduced in the previous example.

• ¿e noncommutative -form iθ is dened as iθ(X + adγ) = γ. It splits the short exact sequenceof Lie algebras and C∞(M)-modules

0 //A //Der(A) //iθ

vv Γ(M) //0 (..)

• Noncommutative integration is a well-dened map of dierential complexes

∫n.c. ∶ ΩDer(A)→ Ω−(n−)(M) ∫n.c. dω = d ∫n.c. ω


Using a metric h onM and the metric gkℓ = n Tr(EkEℓ) on the matrix part, one can dene a

metric on Der(A) as follows,

g(X + adγ,Y + adη) = h(X,Y) + m g(γη)

where m is a positive constant which measures the relative “weight” of the two “spaces”. In physicalnatural units, it has the dimension of a mass.

Consider now the right A-module M = A. As for the algebra Mn, the noncommutative -form−iθ denes a canonical noncommutative connection by the relation ∇−iθ

X a = Xa − iθ(X)a for anya ∈ A.Proposition .. (Properties of ∇−iθ)For any a ∈ A and X = X + adγ ∈ Der(A), one has ∇−iθ

X a = X⋅a − aγ.¿e curvature of the noncommutative connection ∇−iθ is zero.¿e gauge transformed connection ∇−iθg by g ∈ C∞(M) ⊗GLn(C) is associated to the noncom-

mutative -form X↦ −iθ(X) + g−(X⋅g) = −γ + g−(X⋅g).

2.4 The endomorphism algebra of a vector bundle

¿e second example of the previous section mixes together two geometries: the de Rham ordinarydierential geometry, and the noncommutative derivation-based dierential geometry of thematrixalgebra. ¿is last geometry is very similar to the ordinary geometry of the Lie group SLn(C).

It is common in physics to consider the geometry of a based manifold with the geometry of aLie group (especially Lie groups of the type SU(n)): indeed, this is the geometry underlying gaugetheories as they are used in the Standard Model of particle physics. ¿is kind of geometry is wellunderstood in the context of principal ber bundles (see Section .).

¿is section is devoted to the denition of a noncommutative geometry which generalizes andcontains in a precise meaning (see Section .) some essential aspects of the ordinary geometry ofSU(n)-principal ber bundles.

2.4.1 The algebra and its derivationsLet E be a SU(n)-vector bundle overM with ber Cn. Consider End(E), the ber bundle of endo-morphisms of E (see Example ..). We denote by A the algebra of sections of End(E). ¿is is thealgebra we will study using noncommutative dierential geometry.

For later references, the trivial case is the situation where E =M ×Cn is the trivial ber bundle.In that case, one has A = C∞(M) ⊗Mn. Its noncommutative geometry is the one exposed as thesecond example of the previous section. In general, A is (globally) more complicated.

Let us motivate the importance of this algebra by the following remarks:

Remark .. (Relation to ordinary geometry)¿e endomorphism ber bundle End(E) is associated to a SU(n)-principal ber bundle P for thecouple (Mn,Ad).

Because G = SU(n) ⊂ Mn(C) and g = su(n) ⊂ Mn(C), one has P ×α G ⊂ End(E) and AdP =P ×Ad g ⊂ End(E) where αg(h) = g−hg for any g,h ∈ G.

¿is implies that the gauge group G = Γ(P ×αG) and its Lie algebra LieG = Γ(AdP) (see Exam-ple ..) are subspaces of A.


Wewill see in the following that (ordinary) connections are also related to this noncommutativegeometry. ⧫

Locally, using trivialisations of E , the algebra A looks like C∞(U) ⊗Mn. ¿is is very useful tostudy some objects dened on A.

Proposition .. (Basic properties)One has Z(A) = C∞(M).

Involution, trace map and determinant (Tr,det ∶ A→ C∞(M)), are well dened berwise.Let us dene SU(A) as the unitaries inA of determinant , and su(A) as the traceless antihermitean

elements. ¿en G = SU(A) and LieG = su(A).

¿is identies exactly the gauge group and its Lie algebra as natural and canonical subspaces ofA.

Let ρ ∶ Der(A)→ Der(A)/Int(A) = Out(A) be the projection of the short exact sequence (..).¿is projection has an natural interpretation in this context:

Proposition .. (¿e derivations of A)One has Out(A) ≃ Der(C∞(M)) = Γ(M) and ρ is the restriction of derivations X ∈ Der(A) toZ(A) = C∞(M). Int(A) is isomorphic to A, the traceless elements in A.

¿e short exact sequence of Lie algebras and C∞(M)-modules of derivations looks like

0 // Int(A) // Der(A) ρ // Γ(M) // 0X // X

Real inner derivations are given by the adξ with ξ ∈ LieG = su(A).

¿e short exact sequence in this proposition describes the general situation which generalisesthe splitting for the trivial situation encountered in (..). ¿ere is no a priori canonical splitting inthe non trivial case. Moreover, the noncommutative -form iθ is no more dened here. But one candene a map of C∞(M)-modules:

iθ ∶ Int(A)→ A adγ ↦ γ − n Tr(γ)1

Here is an important result which can be proved using local trivialisations:

Proposition ..

ΩDer(A) = Ω

Der(A)

¿e next proposition will be used in the study of ordinary connections on E and their relationsto the noncommutative geometry of A:

Proposition .. (Horizontal forms for the operation of Int(A))¿e space of sections Γ(⋀T∗M⊗End(E)) is the graded algebra of noncommutative horizontal formsin Ω

Der(A) for the operation of Int(A) on ΩDer(A).

2.4 The endomorphism algebra of a vector bundle 81

2.4.2 Ordinary connectionsLet us now show how this noncommutative geometry is well adapted to not only study ordinaryconnections on the vector bundle E , but also,as will be seen in the next section, to allow to somenatural generalisations of these connections.

Let ∇E be any (usual) connection on the vector bundle E . One can dene the two associatedconnections ∇E∗ on E∗ and ∇ on End(E) by the relations

X⋅⟨φ, s⟩ = ⟨∇E∗X φ, s⟩ + ⟨φ,∇EXs⟩ ∇X(φ⊗ s) = (∇E∗X φ)⊗ s + φ⊗ (∇EXs)

with X ∈ Γ(M), φ ∈ Γ(E∗) and s ∈ Γ(E)In the following, we will use the notation X = ρ(X) ∈ Γ(M) for any X ∈ Der(A).

Proposition .. (¿e noncommutative -form α)For any X ∈ Γ(M), ∇X is a derivation of A.

For any X ∈ Der(A), the dierence X − ∇X is an inner derivation. ¿is permits one to introduceX ↦ α(X) = −iθ(X − ∇X). By construction, α is a noncommutative -form α ∈ Ω

Der(A) which givesthe decomposition

X = ∇X − adα(X)

For any γ ∈ A, one has α(adγ) = −γ, for any X ∈ Der(A), one has Trα(X) = , and for anyX ∈ DerR(A) one has α(X)∗ + α(X) = .

Notice that by the decomposition given in this proposition, X ↦ ∇X is a splitting as C∞(M)-modules of the short exact sequence

0 //A //Der(A) //Γ(M) //∇ss

0 (..)

¿e obstruction to be a splitting of Lie algebras is nothing but the curvature of ∇ which we denoteby R(X,Y) = [∇X,∇Y] −∇[X,Y].

Remark .. (α extends −iθ)¿e relation α(adγ) = −γ shows that α extends −iθ ∶ Int(A) → A. As will be seen in Proposi-tion .., any such extension is indeed related to a choice of an ordinary connection of E . ⧫

One can then introduce the main result which connects the ordinary geometry of E and thenoncommutative dierential geometry of A:

Proposition .. (Ordinary connections and noncommutative forms)¿e map ∇E ↦ α is an isomorphism between the ane spaces of SU(n)-connections on E and thetraceless antihermitean noncommutative -forms on A such that α(adγ) = −γ.

¿e noncommutative -form (X,Y) ↦ Ω(X,Y) = dα(X,Y) + [α(X),α(Y)] depends only onthe projections X and Y ofX andY. ¿is means that it is a horizontal noncommutative -form for theoperation of Int(A) on Ω

Der(A).¿e curvature RE of ∇E , considered as a section of ⋀T∗M ⊗ AdP ⊂ ⋀T∗M ⊗ End(E) (see

Proposition ..), is exactly the horizontal noncommutative -form Ω.


Remark .. (¿e intermediate construction in ordinary geometry)We saw in Remark .. that in the ordinary geometry of a principal ber bundle, one is used tointroduce connections as -forms ω ∈ Ω(P) ⊗ g, with two conditions: vertical normalisation andequivariance. Its curvature is then a -form in Ω(P) ⊗ g, equivariant and horizontal. ¿e otherpossibility is to introduce a family of local -formsA ∈ Ω(U)⊗g on open subsetsU of trivialisationsof P , with some non homogeneous gluing relations. ¿e curvature is represented by a family of -forms F ∈ Ω(U)⊗ g satisfying some homogeneous gluing relations.

Using the “top” construction (equivariant and horizontal properties) or the “bottom” one (ho-mogeneous gluing relations), one can show that the curvature is indeed a section of the vector bundle⋀T∗M⊗AdP ⊂ ⋀T∗M⊗ End(E).

¿is proposition shows that this “intermediate” construction (the curvature as a section of a vec-tor bundle) can be completed at the level of the connection -form, at the price of using noncommu-tative geometry (the noncommutative -form α) in order to take into account the non homogeneousgluing relations of the local connection -forms (see Remark ..). ¿e vertical normalisation andthe equivariant conditions at the level of P are replaced by a unique condition on inner derivationsat the level of A. ⧫

Let us now look at gauge transformations. Let u ∈ G = SU(A) and ξ ∈ LieG = su(A).

Proposition .. (Gauge transformations)¿e noncommutative -form αu corresponding to the gauge transformed connection∇Eu is given by thesuggestive expression

αu = u∗αu + u∗du

¿e innitesimal gauge transformation induced by ξ is

α↦ −dξ − [α, ξ] = Ladξα

¿ismeans that we can interpret innitesimal gauge transformations on connections on E as Lie deriva-tive of real inner derivations on A.

Remark .. (Local expressions)It is instructive to look at the noncommutative -form α in some local trivialisation of E . LetUi ⊂Mbe a local trivialisation system of E , and so of End(E). We denote by aloci ∶ Ui → Mn the restrictionof the global section a ∈ A looked at in a local trivialisation.

Over Ui ∩Uj ≠ ∅, one has the homogeneous gluing relations alocj = Adg−ij aloci = g−i j aloci gi j, with

gi j ∶ Ui ∩Uj→ SU(n) the transition functions.Locally a derivationX ∈ Der(A) can be written asXloc

i = Xi + adγi , with γi ∶ Ui →Mn (traceless)and Xi a vector elds on Ui. Using the map ρ, one gets that Xi is the restriction of X = ρ(X) to Ui,so that we can write X = Xi.

Using compatibility with the homogeneous gluing relations for sections, one nds that the γi’ssatisfy some non homogeneous gluing relations

γj = g−i jγigi j+ g−i jX⋅gi j

2.4 The endomorphism algebra of a vector bundle 83

¿e noncommutative -form α is then locally given by the expressions

αloci (X + adγi) = Ai(X) − γi

where theAi’s form the family of local trivialisation of the connection -form. It is then easy to checkthat

αlocj (X + adγj) = Aj(X) − γj = (g−i jAi(X)gi j+ g−i jX⋅gi j) − (g−i jγigi j+ g−i jX⋅gi j)= g−i j (Ai(X) − γi)gi j = g−i jαloci (X + adγi)gi j

so that these expressions indeed dene a global section in A.As can be noticed here, the global existence of the noncommutative -form α relies on the fact

that the Ai’s and the γi’s share the same non homogeneous gluing relations. ⧫

2.5 Noncommutative connections on AIn this section, we study noncommutative connections on the rightA-moduleM = A equipped withthe canonical Hermitean structure (a,b)↦ a∗b.

2.5.1 Main properties

As we saw in Proposition .., a noncommutative connection ∇ on the right A-module M = A isgiven by a noncommutative -form ω ∈ Ω

Der(A) by the relation ∇Xa = Xa + ω(X)a. ¿is impliesthat studying ∇ is equivalent to studying ω.

Let us rst look at some particular noncommutative connections:

Proposition .. (¿e noncommutative connection associated to α)Let∇E be a SU(n)-connection on E , and denote by α its associated noncommutative -form. ¿en, thenoncommutative connection ∇α dened by the noncommutative -form α is given by

∇αXa = ∇Xa + aα(X) (..)

In particular, for any X ∈ Γ(M), one has ∇α∇Xa = ∇Xa.

¿is noncommutative connection ∇α is compatible with the canonical Hermitean structure.¿e curvature of ∇α is Rα(X,Y) = RE(X,Y).A gauge transformation induced by u ∈ G = SU(A) on the connection ∇E induces a (noncommu-

tative) gauge transformation on ∇α.

Proof Recall that by denition, one has X = ∇X − adα(X) and ∇αXa = Xa + α(X)a. ¿is proves

(..).One the other hand, the curvature of ∇α is the noncommutative -form dα(X,Y)+[α(X),α(Y)]

which has been identied with the curvature of ∇ in Proposition ...In a gauge transformation, one has αu = u∗αu + u∗du, which is also the noncommutative gauge

transformation applied to ∇α. ∎


We now arrive at the main result of this report:¿eorem .. (Ordinary connections as noncommutative connections)¿e space of noncommutative connections on the right A-module A compatible with the Hermiteanstructure (a,b)↦ a∗b contains the space of ordinary SU(n)-connections on E .

¿is inclusion is compatible with the corresponding denitions of curvature and gauge transforma-tions.

From now on, one can consider that an ordinary connection is a noncommutative connectionon the right A-module A. In this respect, this point of view generalizes the notion of connectionthrough the intermediate construction.

A natural question is: what are noncommutative connections from a physical point of view?

2.5.2 Decomposition of noncommutative connections on the module AIn order to answer the above question, one can look at some natural decompositions of noncommu-tative connections, and compare these decompositions to “ordinary” connections.

Let us x a connection ∇E on E , and denote by α its associated noncommutative -form. ¿enany noncommutative connection ∇ can be decomposed as

∇Xa = ∇αXa +A(X)a

withA ∈ ΩDer(A), so that ω = α +A is the noncommutative -form for ∇.

Using the relation X = ∇X − adα(X), one splits A as A(X) = a(X) − b(α(X)), where b ∶ A → Ais dened by b(γ) = A(adγ).

A straightforward computation shows that the curvature of ∇ can then be written as

R(X,Y) = RE(X,Y) +∇XA(Y) −∇YA(X) −A([X,Y]) + [A(X),A(Y)]= RE ,a(X,Y) −∇a

Xb(α(Y)) +∇aYb(α(X))

+ [b(α(X)),b(α(Y))] + b(α([X,Y]))

where RE ,a is the curvature of the connection ∇E ,aX s = ∇EXs + a(X)s on E and ∇a is its associatedconnection on End(E).

Performing a gauge transformation with u ∈ G = SU(A), one has

Au = u∗Au + u∗(∇u) au = u∗au + u∗(∇u) bu = u∗bu

Notice the replacement of the dierential by ∇ in these expressions.

Remark .. (Local expressions)In Remark .., we looked at local expressions of the noncommutative -form α. Let us now look atthe previous decomposition in a local trivialisation of E . ¿e noncommutative connection ∇ takesthe local expression:

∇locXlocaloc = X⋅aloc + [A(X) + aloc(X) − bloc(A(X))] aloc + bloc(γ)aloc − alocγ

where A is the local connection -form of ∇E and Xloc = X + adγ as before.In a change of local trivialisation, the two local maps γ↦ bloc(γ) and X↦ aloc(X) transform as

γ↦ b′loc(γ) = g−bloc(gγg−)g X↦ a′loc(X) = g−aloc(X)g

which are both homogeneous gluing relations. ⧫

2.5 Noncommutative connections on A 85

In order to bemore explicit, consider now the trivial situation E =M×Cn andA = C∞(M)⊗Mn.As a reference (ordinary) connection, one can take∇EXs = X⋅s, so that, using the local expression

of α, one hasα(X) = α(X + adγ) = −γ = −iθ(X)

with Trγ = . ¿en ∇α = ∇−iθ. Moreover, b(α(X)) = b(−γ) = −b(γ), so that

∇Xa = X⋅a + a(X)a + b(γ)a − aγ = ∇aXa + b(γ)a − aγ

where ∇a, dened in some local trivialization by ∇aXa = X⋅a+ a(X)a, is an ordinary connection on

End(E), but is not ∇a, which takes the explicit local form ∇aXa = X⋅a + [a(X),a].

X ↦ a(X) behaves like a gauge potential with respect to gauge transformations (here ∇ = d).¿e dierence between ordinary connections and noncommutative connections is the presence ofb, which represents some additional elds in physics. ¿ese elds have homogeneous gauge trans-formations.

¿e curvature can be written, for X = X + adγ andY = Y + adη,

R(X,Y) = RE ,a(X,Y) + (∇aXb)(η) − (∇a

Yb)(γ) + [b(γ),b(η)] − b([γ,η])

where ∇a is the connection (∇aXb)(η) = X⋅b(η) − b(X⋅η) + [a(X),b(η)] on the space of C∞(M)-

linear maps A → A.

2.5.3 Yang-Mills-Higgs Lagrangian on the module AConsider, as before, the trivial case A = C∞(M) ⊗Mn and the right A-module A. Let a = aµdxµand b = bkθk, with aµ,bk ∈ C∞(M)⊗Mn.

¿e curvature is then the noncommutative -form

R = (∂µaν − ∂νaµ + [aµ,aν])dxµdxν + (∂µbk + [aµ,bk])dxµθk +

([bk,bℓ] − Cmkℓbm)θkθℓ

Using a metric (here euclidean) on Der(A) and an associated Hodge star operation, one candene a Lagrangian. Using ordinary and noncommutative integration, one then denes the action:

S(R) = ∫ dxTr∑µ,ν

(∂µaν − ∂νaµ + [aµ,aν])

+m∑µ,k

(∂µbk + [aµ,bk]) +m∑k,ℓ

([bk,bℓ] − Cm

kℓbm)

¿e integrand is zero when

a gauge equivalent to db = [bk,bℓ] = Cmkℓbm

so that the bk’s are constant and induce a representation of sln inMn.For the right A-moduleM = C∞(M)⊗Mr,n, one would get similar results: at connections are

classied by inequivalent representations of sln inMr.

Remark .. (Physical interpretation)From a elds theory point of view, one can notice that the aµ elds behave like ordinary Yang-Millselds, for a SU(n) gauge theory. On the other hand, the interesting point is that the bk elds behave


as Higgs elds: in the above action, the vacuum states can be non trivial and the Higgs mechanismof mass generation is possible. Finally, the coupling between these elds is a covariant derivative inthe adjoint representation. ⧫

For a more general situation where A is not the trivial case, one can proceed in the same line:

• One has to use a reference connection on E to help to decompose noncommutative connec-tions.

• ¿e curvature looks similar except for the presence of the reference connection.

• ¿eHodge star operator is dened.

• ¿e action splits into three terms, and the vacuum states are related to the global structure ofthe vector ber bundle E .

2.6 Relations with the principal fiber bundle

It is possible to look at the noncommutative geometry of A using the ordinary geometry of theunderlying SU(n)-principal ber bundle P and the noncommutative geometry of a bigger algebra,herea er denoted by B.

2.6.1 The algebra BAs before, let P be the SU(n)-principal ber bundle to which E is associated, and consider the as-sociative algebra B = C∞(P) ⊗Mn. ¿is algebra is an example of the trivial situation mentionedin .., so that one has immediately the following facts: the center of B is Z(B) = C∞(P), its Liealgebra and Z(B)-module of derivations splits, Der(B) = Γ(P)⊕ [C∞(P)⊗ sln], and its noncom-mutative dierential calculus is the tensor product of the two dierential calculi associated to P andMn: Ω

Der(B) = Ω(P)⊗ΩDer(Mn) with the dierential d = d + d′.

One can embed the real Lie algebra su(n) as a subalgebra of Der(B) in two ways:

ξ↦ ξv vertical vector eld on P ξ↦ adξ inner derivation

¿is permits one to introduce the following two Lie subalgebras of Der(B):

gad = adξ / ξ ∈ su(n) gequ = ξv + adξ / ξ ∈ su(n)

Proposition ..¿e algebra C∞(P) (resp. A) is the set of invariant elements for the action of gad (resp. gequ) on B.

Proof C∞(P) is the invariants of gad because adξb = for any ξ ∈ su(n) implies b ∈ Z(B). A isthe invariants of gequ because A is the set of sections of End(E), which is FSU(n)(P ,Mn), the spaceof SU(n)-equivariant maps from P to Mn. ¿e relation ξv⋅b + adξb = for any ξ ∈ su(n) is theinnitesimal version of this equivariance. ∎

¿e two Lie subalgebras gad and gequ dene Cartan operations on (ΩDer(B), d). ¿e previous

proposition tells us that the algebras B, C∞(P) and A are related by these two operations.Moreover, C∞(M) is itself the set of invariant elements for ξ↦ ξv in C∞(P) and the invariants

in A for the operation of Int(A).

2.6 Relations with the principal fiber bundle 87

Proposition .. (Relations between the dierential calculi)At the level of dierential calculi, all these relations generalize in the following structure:

Ω(P)⊗ΩDer(Mn) Ω(P)? _basic elements

su(n) ∋ ξ↦ adξoo

ΩDer(A)?

basic elementssu(n) ∋ ξ↦ ξv + adξ

OO

Ω(M)?

basic elementssu(n) ∋ ξ↦ ξv

OO

? _basic elementsInt(A)

oo

In order to show these relations, one need the concept of noncommutative quotient manifoldintroduced in [Masson, ]. We refer to [Masson, ] for the complete proof.

Notice that this proposition contains awell known result in ordinary dierential geometry, whichsays that the space of tensorial forms inΩ(P)⊗g (horizontal and equivariant for the action inducedby right multiplication on P and the adjoint action on the Lie algebra g) is the space Ω(M,AdP)of forms on the base manifoldM with values in the vector bundle AdP . ¿is result permits one toindentify the curvature of a connection on P to a form in Ω(M,AdP) (see Proposition .. andRemark ..).

In Proposition .., we saw that the noncommutative integration is well dened on algebraslike B. ¿is induces a map

∫n.c. ∶ ΩrDer(B)→ Ωr−(n−)(P)

which has the following properties:

Proposition .. (Noncommutative integration)If ω ∈ Ωr

Der(B) is a horizontal (resp. basic) noncommutative form for one of the operations of gad orgequ, then ∫n.c. ω ∈ Ωr−(n−)(P) is horizontal (resp. basic) for the corresponding operation restricted toΩ(P) ⊂ Ω

Der(B).¿is noncommutative integration then restricts to a “noncommutative integration along the non-

commutative ber” ΩDer(A)→ Ω−(n−)(M).

¿is noncommutative integration is compatible with the dierentials, and it induces maps in coho-mologies

∫n.c. ∶ H(ΩDer(B), d)→ H−(n−)

dR (P)

∫n.c. ∶ H(ΩDer(A), d)→ H−(n−)

dR (M)

¿is situation looks very similar to the integration along the bers of compactly supported (alongthe bers) dierential forms in the theory of vector bundles.

2.6.2 Ordinary vs. noncommutative connections

It is instructive to identify ordinary connections in this setting. Let ∇ be a noncommutative connec-tion on the right A-module A, and denote by α ∈ Ω

Der(A) it associated noncommutative -form.As a basic noncommutative -form in Ω

Der(B) for the operation of gequ, one can write

α = ω− ϕ ∈ [Ω(P)⊗Mn]⊕ [C∞(P)⊗Mn ⊗ sl∗n]


0

0

0

// ZDer(A)

// Γ(VP)

// 0

0 // Int(A)

// NDer(A)τ

ρP // ΓM(P)π∗

// 0

0 // Int(A)

// Der(A)

ρ // Γ(M)

// 0

0 0 0

Figure .: Some relations between the derivations of B and A and some vector elds on P andM.

¿e basic condition implies the relations

(Lξv + Ladξ)ω = (Lξv + Ladξ)ϕ = iξvω− iadξϕ =

for any ξ ∈ su(n).Proposition .. (Ordinary connection)Let∇E be an ordinary connection on E and α ∈ Ω

Der(A) its associated noncommutative -form. ¿en,as a basic element in Ω

Der(B), one hasα = ω− iθ

where ω ∈ Ω(P)⊗ su(n) ⊂ Ω(P)⊗Mn is the connection -form on P associated to∇E and iθ is thecanonical noncommutative -form dened in Ω

Der(B) (Proposition ..).

In order to prove this formula, one has to use the equivariance and the vertical condition for ω,and some of the properties listed before on iθ.

Notice that this inclusion of ordinary connection into the space of basic -forms onB is canonical,since the noncommutative -form iθ is itself canonical.

2.6.3 Splittings coming from connections¿eprevious considerations showhow the dierential calculi connect together through someCartanoperations. ¿ere also exist some strong relations between the derivations ofA, some derivations ofB, and some vector elds on P andM. ¿ey are summarised in the diagram of Fig. ..

In this diagram, one has the following short exact sequences of Lie algebras andC∞(M)-modules:

• 0 //Int(A) //Der(A) ρ //Γ(M) //0

¿is is the short exact sequence which relates vector elds onM, derivations on A and innerderivations on A given in Proposition ...

• 0 //ZDer(A) //NDer(A) τ //Der(A) //0

NDer(A) ⊂ Der(B) is the subset of derivations on B which preserve A ⊂ B.ZDer(A) ⊂ Der(B) is the subset of derivations on B which vanish on A. ¿is is a Lie ideal inNDer(A), and τ is the quotient map.¿e Lie algebra ZDer(A) is generated as a C∞(P)-module by the elements ξv + adξ for anyξ ∈ su(n).

2.6 Relations with the principal fiber bundle 89

• 0 //Γ(VP) //ΓM(P) π∗ //Γ(M) //0

¿ese are pure geometrical objects:Γ(VP) is the Lie algebra of vertical vector elds on P .ΓM(P) = X ∈ Γ(P)/π∗X (p) = π∗X (p′) ∀p, p′ ∈ P s.t. π(p) = π(p′) is the Lie algebraof vector elds on P which can be mapped to vector elds onM using the tangent mapsπ∗ ∶ TpP → Tπ(p)M.

• 0 //Int(A) //NDer(A) ρP //ΓM(P) //0

Here, the elements in Int(A) are identied to the adγ for γ ∈ A ⊂ B. Int(A) is then a Liesubalgebra ofNDer(A).ρP is the restriction to NDer(A) of the projection on the rst term in the splitting Der(B) =Γ(P)⊕ [C∞(P)⊗Der(Mn)].

An ordinary connection ω ∈ Ω(P)⊗su(n) splits these short exact sequences. Let us look moreclosely at the central square of the diagram of Fig. .. One can dene splittings as follows:

NDer(A) ρP // //

τ

ΓM(P)

π∗

(π∗X )h + ω(X )v + adω(X ) Xoo

ρ(X)h − adα(X)B Xh

X_

OO

X_

OO

∇X Xoo

Der(A) ρ// // Γ(M)

where:

• Γ(M)→ Der(A), X↦ ∇X:

¿is is the splitting mentioned in Proposition .., which li s vector elds onM into deriva-tions on A.

• Γ(M)→ ΓM(P), X↦ Xh:

¿is splitting li s vector elds onM into horizontal vector elds Xh on P through the ordi-nary geometrical procedure. Using its equivariance, one can easily verify that the vector eldXh is indeed a π∗-projectable vector eld. In fact, for any X ∈ ΓM(P), one has X = (π∗X )h +X v, where X v is the vertical projection of X , explicitly given by the formula X v = ω(X )v.

• Der(A)→ NDer(A), X↦ ρ(X)h − adα(X)B :

¿is li s derivations onA into derivations on B. Here, α(X)B is the basic element in B associ-ated to α(X) ∈ A and ρ(X)h ∈ Γ(P) is the horizontal li of the vector eld ρ(X). By construc-tion, one has adα(X)B ∈ NDer(A). On the other hand, one veries that for any X ∈ Γ(M) andany ξ ∈ su(n), [ξv + adξ,Xh] = (as an element in Der(B)), which shows that Xh ∈ NDer(A).¿e relation τ(ρ(X)h − adα(X)B) = X relies on the two following facts: in the identication ofa ∈ A as an equivariant map aB ∶ P → Mn, one has the identication of ∇Xa with Xh⋅aB, andone has the decomposition X = ∇ρ(X) − adα(X).


• ΓM(P)→ NDer(A), X ↦ (π∗X )h + ω(X )v + adω(X ):Here, we li π∗-projectable vector elds X on P into derivations on B. Notice that for anyX ∈ ΓM(P), one has the decomposition X = (π∗X )h + ω(X )v. ¿e inner derivation adω(X )is there in order that ω(X )v + adω(X ) ∈ NDer(A) (we know from the previous result that(π∗X )h ∈ NDer(A)). In fact, one has the more interesting result that

ω(X )v + adω(X ) ∈ ZDer(A)

In order to better understand the two li ings ending in NDer(A), it is useful to characterizederivations in NDer(A). Such a derivation can be decomposed, as an element in Der(B), as X =X + adb, with X ∈ Γ(P) and b ∈ B = C∞(P) ⊗ sln. Using the fact that ρP is just the restriction ofX to C∞(P), one has ρP(X) = X ∈ ΓM(P). ¿e condition X ∈ NDer(A) implies that [ξv + adξ, X] ∈ZDer(A) for any ξ ∈ su(n). Using the structure ofZDer(A), one canwrite [ξv+adξ, X] = fi(ηvi +adηi)for some fi ∈ C∞(P) and ηi ∈ su(n), which can be decomposed into two parts: [ξv,X ] = fiηvi andξv⋅b + [ξ,b] = fiηi.

Denote by Lequξ = Lξv+adξ the Lie derivative associated to the Cartan operation of the Lie algebragequ on (Ω

Der(B), d). ¿e second relation is then Lequξ b = fiηi. Applying now the connection -formω on the rst relation, one gets ω([ξv,X ]) = fiηi, which can be written, using the equivarianceof ω: Lequξ ω(X ) = fiηi. ¿e dierence a(X) = ω(X ) − b is then Lequ-invariant, which means thata(X) ∈ A. With X = τ(X), this is exactly the element α(X) ∈ A identied as an element in B, whereα is the noncommutative -form associated to the connection ω.

Using these constructions, one has the following decomposition of any X ∈ NDer(A):

X = X + adb = (π∗X )h + ω(X )v + adω(X )´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

∈ZDer(A)

− ada(X)²∈Int(A)

2.7 Cohomology and characteristic classes

In ordinary dierential geometry, it is possible to relate the cohomology of a ber bundle to thecohomology of its base manifold using a spectral sequence based on a Čech-de Rham bicomplexconstructed using dierential forms. We will show that such a construction can be performed withthe space of noncommutative dierential forms.

Using the noncommutative geometry structures described above, it is also possible to recoverthe Chern characteristic classes of the vector bundle E . ¿e construction we present in the followingis purely algebraic, and relies on an adaptation of some work by Lecomte about characteristic classesassociated to splitting of short exact sequence of Lie algebras.

2.7.1 The cohomology of ΩDer(A)Let us recall the Leray theorem in ordinary dierential geometry.

¿eorem .. (Leray)For any ber bundle F //E π //M , there exists a spectral sequence Er converging to the cohomologyof the total space H

dR(E) withEp,q = Hp(U;Hq)

2.7 Cohomology and characteristic classes 91

⋮ ⋮ ⋮ . . . ⋮ . . .

0 // Ωq(M) δ //dOO

∏Ωq(Uα)δ //

dOO

∏Ωq(Uαα)δ //

dOO

. . . δ //∏Ωq(Uα...αp)δ //

dOO

. . .

⋮ ⋮dOO

⋮dOO

⋮dOO

. . . ⋮dOO

. . .

0 // Ω(M)dOO

δ //∏Ω(Uα)dOO

δ //∏Ω(Uαα)dOO

δ // . . . δ //∏Ω(Uα...αp)dOO

δ // . . .

0 // Ω(M)dOO

δ //∏Ω(Uα)dOO


δ // . . . δ //∏Ω(Uα...αp)dOO

δ // . . .

0 // Ω(M)dOO

δ //∏Ω(Uα)dOO


δ // . . . δ //∏Ω(Uα...αp)dOO

δ // . . .

C(U;R)iOO

δ // C(U;R)iOO

δ // . . . δ // Cp(U;R)iOO

δ // . . .

0

OO

0

OO

. . . 0

OO

Figure .: ¿e ordinary Čech-de Rham bicomplex associated to a ber bundle F //E π //M

whereHq(U) = HqdR(π−U) is a locally constant presheaf on the good covering U ofM.

IfM is simply connected and HqdR(F) is nite dimensional, then

Ep,q = HpdR(M)⊗Hq

dR(F)

One of the proofs of this theorem relies on the construction of a Čech-de Rham bicomplex asillustrated in the diagram of Fig. . (see [Bott and Tu, ] for instance):

Kp,q = ∏α<⋅⋅⋅<αp

Ωq(EUα ...αp) = ∏α<⋅⋅⋅<αp

Ωq(π−Uα...αp)

with Uα...αp = Uα ∩ ⋯ ∩ Uαp for Uαi ∈ U, where U is a good cover ofM, d ∶ Kp,q → Kp,q+ isthe ordinary de Rham dierential on the spaces Ω(EUα ...αp), and δ ∶ Kp,q → Kp+,q is the Čechdierential

(δωp)α...αp+ =p+

∑i=

(−)iωα...αi...αp+ ∣Uα ...αp+

One can introduce a noncommutative Čech-de Rham bicomplex for A. In order to do that,denote by A(U) ≃ C∞(U)⊗Mn the sections of End(E) restricted over a local trivialisationU ⊂MwithU ∈ U, where as before U is a good cover ofM. Denote by gUV ∶ U ∩V → SU(n) the transitionfunctions for E .

For any noncommutative p-form ω = ada⋯dap ∈ ΩpDer(A(U)) and any dierential function

g ∶ U → SU(n), dene the action of g on ω by ωg = (g−ag)d(g−ag)⋯d(g−apg).

Lemma .. (¿e presheafΩDer(A(U)))

For any V ⊂ U, the maps iVU ∶ ΩDer(A(U)) → Ω

Der(A(V)) given by ω ↦ (ω∣V)gUV (restriction to Vand action of gUV) give to U ↦ Ω

Der(A(U)) a structure of presheaf onM, which we denote by F .


Using this presheaf, one can introduce the bicomplex

Cp,q(U;F) = ∏α<⋯<αp

ΩqDer(A(Uα...αp))

where by convention the trivialisation over Uα...αp is the one over Uαp. Let d ∶ Cp,q → Cp,q+ be thenoncommutative dierential, and dene δ ∶ Cp,q(U;F)→ Cp+,q(U;F) by (here gαβ = gUαUβ)

(δω)α...αp+ =p

∑i=

(−)i(ωα...αi...αp+)∣Uα ...αp+ + (−)p+(ωα...αp)gαpαp+∣Uα ...αp+

Notice that in the last term, the action of gαpαp+ performs the change of trivialisation from the oneabove Uαp to the one above Uαp+ .

Denote by C−,q(U;F) = ΩqDer(A) and dene δ ∶ C−,q(U;F) → C,q(U;F) as the restrictions to

the trivialisations of the good cover.One has the following results about the cohomology of Ω

Der(A):

¿eorem .. (Noncommutative Leray theorem)¿e cohomology of the total complex of the bicomplex (C,(U;F), d,δ) is the cohomology ofΩ

Der(A).¿e spectral sequence Er associated to the ltration

FpC(U;F) =⊕s≥p⊕q≥Cs,q(U;F)

converges to the cohomology of ΩDer(A) and satises

E = HdR(M)⊗ I(⋀sl∗n)

Recall that the structure of I(⋀sl∗n) is known. One can nd the proof of this result in [Masson,].

2.7.2 Characteristic classes and short exact sequences of Lie algebrasLet us now show that the splitting (..) of the short exact sequence of derivations contains all theinformations needed to recover the Chern characteristic classes of the ber bundle E . In order to dothat, one has rst to introduce a construction by Lecomte (see [Lecomte, ]).

Let 0 //i //gπ //h //0 be a short exact sequence of Lie algebras, and let φ ∶ h → g be a

morphism which splits it as vector spaces. Dene Rφ = dhφ + [φ,φ] ∶ ⋀h∗ ⊗ g with dh the

dierential on ⋀h∗ ⊗ g for the trivial representation of h on g. For any x, y ∈ h, the quantityRφ(x, y) = −φ([x, y])+ [φ(x),φ(y)] is exactly the obstruction on φ to be a Lie algebra morphism,i.e. a splitting of Lie algebras.

It is worth to note that Rφ looks like a curvature, and indeed, the following construction treatsit as if it were a curvature. One can show that Rφ belongs to ⋀h∗ ⊗ i and that it satises a Bianchiidentity dhRφ + [φ,Rφ] = .

Now, let V be a vector space and ρ a representation of h on V. Denote by Sqρ(i,V) the space oflinear symmetric maps ⊗qi → V which intertwine the adjoint representation ad⊗

qof g on ⊗qi and

the representation ρ π of g on V. Let є be the antisymmetrisation map ⊗h∗ → ⋀h∗.One has the following result, shown in [Lecomte, ]:

2.7 Cohomology and characteristic classes 93

Proposition .. (Characteristic classes of a short exact sequence of Lie algebras)For any α ∈ Sqρ(i,V), let αφ = є α(Rφ ⊗⋯⊗ Rφ) ∈ ⋀qh∗ ⊗V. ¿en one has dαφ = where d is thedierential of the complex ⋀h∗ ⊗V.

¿e cohomology class of αφ in Hq(h;V) does not depends on the choice of φ.If the short exact sequence is split exact as a Lie algebra short exact sequence then this cohomology

class is zero.

Let us adapt this construction to the short exact sequence

0 //Int(A) //Der(A) ρ //Γ(M) //0

It is possible to generalise the previous construction in order to take into account the extra structuresof Z(A)-modules.

We identify Int(A) with A. ¿e adjoint representation of Der(A) on Int(A) is explicitly givenby adX(ada) = [X,ada] = adX(a) so that it is (X,a)↦ X(a) on A.

¿e vector space (and Z(A)-module) we consider is Z(A) itself, on which the representation ρis (X, f)↦ ρ(X)⋅f.

Let SqZ(A)(A,Z(A)) be the space of Z(A)-linear symmetric maps ⊗qZ(A)A → Z(A) which

intertwine the adjoint representation ad⊗qof Der(A) on ⊗q

Z(A)Int(A) = ⊗qZ(A)A and the represen-

tation ρ of Der(A) on Z(A).Notice that, thanks to the Z(A)-linearity, maps in SqZ(A)(A,Z(A)) are local onM, so that

one can look at them in local trivialisations of E . In such a trivialisation over an open set U, theintertwining relations can be written, with the usual notation Xloc = X + adγ:

q

∑i=ϕ(a ⊗⋯⊗ X⋅ai ⊗⋯⊗ aq) = X⋅ϕ(a ⊗⋯⊗ aq)

q

∑i=ϕ(a ⊗⋯⊗ [γ,ai]⊗⋯⊗ aq) =

for any ai ∶ U → sln.Proposition .. (Characteristic classes of E)¿e space SqZ(A)(A,Z(A)) is well dened, which means that theZ(A)-linearity and the intertwiningcondition are compatible, and one has

SqZ(A)(A,Z(A)) = PqI (sln)

the space of invariant polynomials on the Lie algebra sln.¿e dierential complex in which the characteristic classes for the splitting are dened is

HomZ(A)(⋀Z(A)Γ(M),Z(A))

which is the de Rham complex of dierential forms onM.¿e characteristic classes one computes in this way are precisely the ordinary Chern characteristic

classes of the vector bundle E (or of the principal ber bundle P).

¿e last statement relies on the fact that any ordinary connection∇E on E gives rise to a splittingof the short exact sequence whose curvature is exactly the obstruction to be a morphism of Liealgebras. ¿e construction based on SqZ(A)(A,Z(A)) = PqI (sln) is then the ordinary Chern-Weilmorphism.


2.8 Invariant noncommutative connections

Manyworks have been done in the theory of ordinary connections which are symmetric with respectto the action of a Lie group. ¿is leads to understand some ansatz used to get exact solutions of Yang-Mills theories, and recover or introduce some Yang-Mills models coupled with scalar elds throughthese symmetric reductions.

In this section, we generalize these considerations to noncommutative connections on the alge-bra A, and show that the geometrico-algebraic structures introduced so far are very natural in thetheory of symmetric reductions.

¿is exposé is based on [Masson and Sérié, ], and we refer to this paper for more details andreferences.

2.8.1 Action of a Lie group on a principal fiber bundleLet us recall some general constructions that were introduced in the theory of symmetric reductionof connections.

Let G //P π //M be aG-principal ber bundle, and letH be a Lie group acting on the le on P ,such that the action commutes with the right action of G.

¿en, the action of H on P induces a le action of H onM. In the following, we assume thatthis action is simple, which means thatM admits the ber bundle structure H/H //M //M/Hwhere H is an isotropy subgroup: H = Hx = h ∈ H / h⋅x = x. In particular, all the isotropysubgroups are isomorphics to one of them. We x H as such an isotropy subgroup.

¿en we introduce the following spaces:

• N = x ∈M / Hx = H is the space of points inM whose isotropy subgroup is exactly H.

• NH(H) = h ∈ H / hH = Hh is the normalizer of H in H.

• H is a normal subgroup ofNH(H), and one has the principal ber bundle

NH(H)/H //N //M/H

¿e ber bundle H/H //M //M/H is associated to this bundle for the natural action ofNH(H)/H on H/H by (right) multiplication.

Dene S = H×G. ¿is group acts on the right on P by the following relation: (h, g)⋅p= h−⋅p⋅g.For any p ∈ P , let λp ∶ Hπ(p) → G be dened such that h⋅p = p⋅λp(h). ¿en one can show thefollowings:

• Sp = (h, λp(h)) / h ∈ Hπ(p) is the isotropy subgroup of p ∈ P for the action of S. ¿isimplies that the action of S on P is simple.

• Fix an isotropy subgroup S and letQ = p ∈ P / Sp = S. ¿en

S/S //P //M/H is associated to NS(S)/S //Q //M/H

as for the (simple) action of H onM.

2.8 Invariant noncommutative connections 95

ZG(λ(H)) //

n

NS(S)/S // // o

(NS(S)/S)/ZG(λ(H)) w

**

NH(H)/H s&&

G // S/S // //

H/H

ZG(λ(H)) // o

QπQ // //

p

!!

π(Q) x

**

N t

''

G // P π // //

M

M/H M/H

M/H

M/H M/H

Figure .: In this diagram, some arrows represent true applications and other arrows are part ofdiagrams of brations, most of them explicitly given before. Some horizontal arrows correspond tothe action ofG (or subgroups ofG) and some vertical arrows correspond to actions of groups relatedto H and S.

Proposition .. (Some properties ofQ and λ)¿e map λp ∶ Hπ(p) → G such that h⋅p= p⋅λp(h) satises

λp⋅g(h) = g−λp(h)g

For any q ∈ Q, λq depends only on π(q) ∈M: λq(h) = λq⋅g(h). For a xed x inMwhose isotropygroup is H, we denote this map restricted toQ by λ ∶ H → G.

¿e projection π ∶ P →M induces the ber bundle structure

ZG(λ(H)) //Q πQ //π(Q)

with ZG(λ(H)) = g ∈ G / gλ(h) = λ(h)g,∀h ∈ H, the centralizer of λ(H) in G, andπ(Q) ⊂ N .

We summarize in Fig. . all the brations one can obtain relating the spaces introduced so far.In the following, we will concentrate more precisely on the diagram of brations:

NS(S)/S _

// Q _

//M/H

S/S // P //M/H

In order to connect this construction to some dierential structures, we are now interested inthe Lie algebras of the dierent groups introduced before.


Let us look at the structure of the Lie algebra h of the group H. One can introduce the followingspaces:

• h is the Lie algebra of H, the once for all xed isotropy group.

• k is the Lie algebra of the quotient groupNH(H)/H.

• n = h ⊕ k is the Lie algebra ofNH(H), the normalizer of H in H.

• l is the vector space in the orthogonal decomposition h = n D l such that [n, l] ⊂ l (this iscalled a reductive decomposition of h along n).

Denote by g the Lie algebra of the group G. As before, we can introduce the following spaces:

• z the Lie algebra of ZG(λ(H)), the centralizer of λ(H) in G.

• m the vector space in the orthogonal and reductive decomposition g = z D m ([z,m] ⊂ m).

¿e Lie algebra of the group S = H ×G is s = h⊕ g, and one has

• s = (x, λ∗x) / x ∈ h is the Lie algebra of S, the xed isotropy group.

• s ⊕ k⊕ z is the Lie algebra ofNS(S), the normalizer of S in S.

• k⊕ z is the Lie algebra of the quotient groupNS(S)/S.

With these spaces, one has the following result:

Proposition .. (Decomposition of TP)For any q ∈ Q, one has kQq ⊕ z

Qq ⊂ TqQ and TqP = TqQ⊕ lPq ⊕mPq .

In this proposition, we use the following compact notation: aRq is the space of tangent vectorsover q associated to elements x ∈ a ⊂ h or g through the fundamental vector elds on R = Q or Pfor the action of the corresponding group H or G.

2.8.2 Invariant noncommutative connectionsIt is now possible to mix together the geometrical constructions of the previous subsection and thenoncommutative algebraic considerations on the endomorphism algebra associated to aG-principalber bundle P with G = SL(n) or G = SU(n). Let then as before H be a compact connected Liegroup acting on P .Proposition .. (Operations of h onΩ

Der(B) andΩDer(A))

¿eoperation of h onΩ(P) induced by the action ofH onP extends to an operation of h onΩDer(B) =

Ω(P) ⊗ ΩDer(Mn) (using a trivial action on the second factor). ¿is operation commutes with the

operations of gad and gequ, and so reduces to the operation of h on Ω(P) and to an operation of h onΩDer(A).

Denition .. (Invariant noncommutative connection)¿e operation of h onΩ

Der(A) obtained in Proposition .. is our denition of the (noncommuta-tive) action of H on the algebra A.

A noncommutative connection ∇ on the right A-module M = A is said to be h-invariant if,∀y ∈ h, ∀X ∈ Der(A) and ∀a ∈ A, one has Ly(∇Xa) = ∇[y,X]a + ∇X(Lya), where Ly is the Liederivative of the operation of h on Ω

Der(A). ⧫


Using this denition, one obtains the equivalent characterization:

Proposition .. (Invariance of the noncommutative -form α)¿e noncommutative connection ∇ is h-invariant if and only if its noncommutative -form α is invari-ant: Lyα = for all y ∈ h.

¿is last proposition, combined with the relations between the noncommutative geometries ofthe algebrasA andB, permits one to reduce the problem of nding the h-invariant noncommutativeconnections ∇ on the rightA-moduleM = A to the following problem: nd all the noncommutative-forms written as α = ω− ϕ ∈ [Ω(P)⊗Mn]⊕ [C∞(P)⊗Mn ⊗ sl∗n] satisfying the four relations

(Lξv + Ladξ)ω = (Lξv + Ladξ)ϕ = iξvω− iadξϕ = Ly(ω− ϕ) =

for all ξ ∈ g = sln and y ∈ h. ¿e three rst relations express the basicity of α for the operation of gequon Ω

Der(B), and the last one is the h-invariance.¿is last relation decomposes into two independent equations Lyω = and Lyϕ = for all y ∈ h.

¿is implies in particular that one can restrict the study ofω and ϕdened overP to the submanifoldQ ⊂ P . For all q ∈ Q, one has then to characterize the maps

ωq ∶ TqP = TqQ⊕ lPq ⊕mPq →Mn ϕq ∶ g→Mn

Now, the relation iξvω − iadξϕ = for all ξ ∈ g, says that ϕq(ξ) is completely determined byωq(ξvq). It is then sucient to study ωq.

Let us rst consider the TqQ part of TqP . Denote by µq ∶ TqQ→Mn the restriction of ωq to TqQ.One has zQq ⊂ TqQ, so that µ and ϕ are both dened on z ⊂ g, where they coincide: µ(zQ) = ϕ(z)for any z ∈ z. Denote by ηq the restriction of ϕq to z, which is then also the restriction of µq to zQq .It is possible to write down these relations in a compact way through the following result:

Proposition .. (¿e algebraW)LetW = ZMn(λ∗g) be the centralizer of λ∗g in Mn. It is an associative algebra and z ⊂ Der(W).Let Ω

z(W) =W⊗⋀z∗ be the restricted derivation-based dierential calculus associated to it.¿ere is a natural operation of z on Ω(Q)⊗Ω

z(W), and µ − η ∈ (Ω(Q)⊗Ωz(W))

z-basic.

In order to take into account the remaining part of µq in TqQ, we introduce the following biggerdierential calculus:

Proposition .. (¿e dierential calculusΩk⊕z(M/H;W))

¿ere are natural operations of k ⊂ h and z ⊂ g on the dierential algebra Ω(Q)⊗Ωz(W)⊗⋀k∗.

DeneΩ

k⊕z(M/H;W) = (Ω(Q)⊗Ωz(W)⊗⋀k∗)

k⊕ z-basic

¿e algebra C = Ωk⊕z(M/H;W) = (C∞(Q)⊗W)k⊕ z-invariants is the algebra of sections of aW-ber

bundle associated to the principal ber bundle

NS(S)/S //Q //M/H

Recall that kQq ⊂ TqQ. For any k ∈ k, let us dene νq(k) = µq(kQq ), so that ν ∈ C∞(Q)⊗W⊗⋀k∗.¿is element contains the dependence of µq over kQq ⊂ TqQ, but ν and µ∣kQ are not equal as elementsin (Ω(Q)⊗Ω

z(W)⊗⋀k∗) (they do not have the same tri-graduation).


Proposition .. (¿e TqQ part of TqP)One has µ − η − ν ∈ Ω

k⊕z(M/H;W), and this expression contains all the information about therestriction of ω to TQ.

Let us now look at the lPq ⊕mPq part of TqP .Recall that [h⊕ k, l] ⊂ l and [z,m] ⊂ m, so that there are natural actions [h, l⊕m] ⊂ l⊕m and

[k⊕ z, l⊕m] ⊂ l⊕m. On the other hand, recall that s⊕ k⊕ z is the Lie algebra ofNS(S) and k⊕ zis the Lie algebra ofNS(S)/S, with s = (x, λ∗x) / x ∈ h.

On the restriction of ω to Q, the H-invariance and the G-invariance combine together into aNS(S)-invariance. One can treat this invariance in two steps: one for S and the other one forNS(S)/S.

In order to encode the S-invariance, let us dene the vector space of S-invariant linear mapsl⊕m→Mn:

F = f ∶ l⊕m→Mn / f([x,v]) − [λ∗x, f(v)] = , ∀x ∈ h, ∀v ∈ l⊕m

on which k ⊕ z acts naturally using the Lie derivative (Lk+z f)(v) = −f([k,v]) + [z, f(v)] for anyk ∈ k and z ∈ z.

¿eNS(S)/S-invariance is then encoded into the spaceM = (C∞(Q)⊗ F)k⊕ z-invariants.

Proposition .. (¿e lPq ⊕mPq part of TqP)M is the space of sections of the F-ber bundle associated to the principal ber bundle

NS(S)/S //Q //M/H

It is a C-bimodule.¿e restriction of ω to the subspaces lPq ⊕mPq is inM.

Using the two previous decomposition of ω, one get the nal identication:

¿eorem .. (¿e space of H-invariant noncommutative connections)¿e space of H-invariant noncommutative connections on the endomorphism algebra A over the rightA-module A is the space Ω

k⊕z(M/H;W)⊕M.

It is important to notice the following facts:

Remark .. (Naturality of the spaces)In this result, all the spaces are constructed from the principal ber bundle

NS(S)/S //Q //M/H

with the help of geometrical or algebraic methods which are natural in this noncommutative frame-work:

• C = (C∞(Q)⊗W)k⊕ z-invariants is modeled on the nite dimensional algebraW ⊂Mn. It lookslike a “reduced algebra” constructed from A, as A itself is a reduced algebra for B.

• Ωk⊕z(M/H;W) is a natural dierential calculus over C.

• M = (C∞(Q)⊗ F)k⊕ z-invariants is a natural C-bimodule.


• ¿e space SU(C) acts naturally on the spaceΩk⊕z(M/H;W)⊕M as restriction of noncom-

mutative gauge transformations.

• All these spaces are sections of ber bundles over the base spaceM/H. ⧫

We refer to [Masson and Sérié, ] for examples of such symmetric noncommutative restric-tions, in particular the noncommutative generalisation of the well studied situation of sphericalSU() gauge elds over a at four dimensional space-time. For purely noncommutative situations,the problem reduces to the study of the decomposition of some representation ofG inMn into irre-ducible representations.


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