UPPSALA DISSERTATIONS IN MATHEMATICS

110

Structure and representations of certainclasses of infinite-dimensional algebras

Brendan Frisk Dubsky

Department of MathematicsUppsala University

UPPSALA 2018

Dissertation presented at Uppsala University to be publicly examined in Häggsalen,Lägerhyddsvägen 1, Uppsala, Wednesday, 5 December 2018 at 13:15 for the degree of Doctorof Philosophy. The examination will be conducted in English. Faculty examiner: ProfessorRolf Farnsteiner (Christian-Albrechts-Universität zu Kiel).

AbstractFrisk Dubsky, B. 2018. Structure and representations of certain classes of infinite-dimensionalalgebras. Uppsala Dissertations in Mathematics 110. 32 pp. Uppsala: Department ofMathematics. ISBN 978-91-506-2728-2.

We study several infinite-dimensional algebras and their representation theory.In Paper I, we study the category O for the (centrally extended) Schrödinger Lie algebra,

which is an analogue of the classical BGG category O. We decompose the category into a directsum of "blocks", and describe Gabriel quivers of these blocks. For the case of non-zero centralcharge, we in addition find the relations of these quivers. Also for the finite-dimensional partof O do we find the Gabriel quiver with relations. These results are then used to determine thecenter of the universal enveloping algebra, the annihilators of Verma modules, and primitiveideals of the universal enveloping algebra which intersect the center of the Schrödinger algebratrivially.

In Paper II, we construct a family of path categories which may be viewed as locally quadraticdual to preprojective algebras. We prove that these path categories are Koszul. This is doneby constructing resolutions of simple modules, that are projective and linear up to arbitraryposition. This is done by using the mapping cone to piece together certain short exact sequenceswhich are chosen so as to fall into three managable families.

In Paper III, we consider the category of injections between finite sets, and also the pathcategory of the Young lattice subject to the relations that two boxes added to the same columnin a Young diagram yields zero. We construct a new and direct proof of the Morita equivalenceof the linearizations of these categories. We also construct linear resolutions of simple modulesof the latter category, and show that it is quadratic dual to its opposite.

In Paper IV, we define a family of algebras using the induction and restriction functorson modules over the dihedral groups. For a wide subfamily, we decompose the algebras intoindecomposable subalgebras, find a basis and relations for each algebra, as well as explicitlydescribe each center.

Keywords: infinite-dimensional algebras, representation theory, category O, koszul, koszulity,injections, dihedral, preprojective, young lattice

Brendan Frisk Dubsky, Department of Mathematics, Box 480, Uppsala University, SE-75106Uppsala, Sweden.

© Brendan Frisk Dubsky 2018

ISSN 1401-2049ISBN 978-91-506-2728-2urn:nbn:se:uu:diva-363403 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-363403)

List of papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I Brendan Dubsky, Rencai Lü, Volodymyr Mazorchuk and KaimingZhao. Category O for the Schrödinger algebra. Linear Algebra and itsApplications, 460:17-50, 2014.

II Brendan Dubsky. Koszulity of some path categories. Communicationsin Algebra, 45(9):4084-4092, 2017.

III Brendan Dubsky. Incidence category of the Young lattice, injectionsbetween finite sets, and Koszulity. Manuscript, 2018.arXiv:1607.00426

IV Brendan Dubsky. Induction and restriction on representations ofdihedral groups. Manuscript, 2018. arXiv:1805.02567

Reprints were made with permission from the publishers.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Associative algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Representations and modules of associative algebras . . . . . . . . . . . . . . . . . . 102.3 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Quiver algebras and path categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Groups and group algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Lie algebras and universal enveloping algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.7 Homological algebra and Koszulity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.8 Induction and restriction of representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.9 Category O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Sammanfattning på svenska (Summary in Swedish) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.1 Populärvetenskaplig introduktion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Sammanfattning av artiklar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2.1 Artikel I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2.2 Artikel II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2.3 Artikel III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2.4 Artikel IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1. Introduction

For most of us (and indeed most probably for humanity itself), the journeyinto the world of mathematics began with a collection of concrete object –say a handful of pebbles – and the operation of adding numbers of themtogether. In a first, tentative leap of abstraction, we realized that the rules ofaddition of natural numbers model addition of numbers of objects irrespectiveof any physical properties those objects might have; only the numbers count1.Natural numbers together with the addition operation form a basic example ofan algebraic structure.

We soon proceeded to consider more operations (subtraction, multiplication,and division) and more abstractions of the physical world to which we can applythem (negative numbers, rational numbers, real numbers, complex numbers,and later matrices of numbers). Early algebra to a large extend revolvedaround the study of equations involving this quite limited number of operations.Over the past two centuries or so, developments both in the study of theseequations and in physics have motivated radically new kinds of operations andabstractions of physical features, and the modern field of algebra comprisesthe study of a plethora of algebraic structures.

In the subdiscipline of representation theory, algebraists consider certainalgebraic structures – so-called representations – each of which subsume struc-tural properties of another algebraic structure of interest. This is typically doneeither because the representations is how that algebraic structure arises in someapplication, or because the representations embody interesting properties ofthe original structure while being easier to study. The most classical andwidespread kind of representation is the one consisting of certain collectionsof complex matrices equipped with the structure of a vector space and theoperation of matrix multiplication (or more generally linear transformationsequipped with function composition). This kind of representation may beused to study many different algebraic structures, including quiver algebras,groups and Lie algebras, and every such representation may be viewed as arepresentation of some associative algebra.

The present thesis is a collection of results on the complex representationtheory of various associative algebras (in paper II viewed as path categories).In paper I, we study the category O of the Schrödinger Lie algebra. In paperII, we consider the representation theoretic property of Koszulity, and provethat path algebras of a certain class are Koszul. In paper III, we derive a more

1No pun intended.

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elementary and explicit proof of Koszulity as well as a description of the quiverof the algebra of injections of finite sets and a proof of its Koszul self-duality.Finally, in paper IV, we study certain algebras generated by the induction andrestriction functors on representations of dihedral groups.

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2. Preliminaries

Here we introduce some of the main concepts used in the papers to follow.While a solid background in mathematics will be necessary to understand thecontent of the papers, the present chapter is intended to provide an accessiblereference for the mathematician whose algebra is a bit rusty, as well as arelatively self-contained overview of the studied areas for the mathematicallyinterested general audience. References and directions for further reading onthe various topics are given at the end of each section. All definitions will begiven for the context of the complex numbers, C, as this is the predominantcase in the papers, but will typically have straightforward analogues for otherfields.

2.1 Associative algebrasAn associative algebra, A, is a vector space equipped with a bilinear andassociative multiplication operation, i.e. a binary operation _·_ (multiplicationsymbols such as this dot are often omitted in favor of mere juxtaposition) whichsatisfies for any a, b, c, d ∈ A and k ∈ C the following.(i) (a+ b) · (c+ d) = a · c+ a · d+ b · c+ b · d,(ii) a · (b · c) = (a · b) · c,(iii) k(a · b) = (ka) · b = a · (kb).

An associative algebra A is said to be finite-dimensional if A is finite-dimensional as a vector space, and unital if there is a unit element in A, i.e. anelement 1 ∈ A such that 1 · a = a = a · 1 for any a ∈ A. Associative algebrasare often tacitly assumed to be unital as part of the definition. Paper II studieslinear path categories with infinitely many objects, and although these may infact equivalently be described as (non-unital) associative algebras, we speak ofthem in the language of category theory in order to emphasize the differencefrom unital associative algebras.

Example 2.1.1. For any integer n ≥ 1, the vector space of n × n-matriceswith matrix multiplication as the operation _ · _ is a finite-dimensional unitalassociative algebra with the n× n identity matrix as unit.

The following example generalizes the previous one by the usual identifica-tion of matrices with linear maps.

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Example 2.1.2. Let V be an arbitrary vector space. Then the vector spaceEnd(V ) with function composition as _ · _ is a unital associative algebra withthe identity map as unit.

For further reading on associative algebras, see [1].

2.2 Representations and modules of associative algebrasLet A be an associative algebra. A representation of A is an algebra homo-morphism

ϕ : A → End(V ),

where End(V ) is the endomorphism algebra of a vector space V , as in Example2.1.2. The property that there is a homomorphism from A to End(V ) shouldbe intuitively understood to mean that some of the algebraic structure of A isto be found also in End(V ).

A vector space V is said to be a (left) A-module if it is equipped with abilinear A-action _ ∗ _ such that

(a · b) ∗ v = a ∗ (b ∗ v)for any a, b ∈ A and v ∈ V and in the case of unital algebras furthermoresuch that 1 ∗ v = v. Again, the operation symbol ∗ is often omitted infavor of juxtaposition. It can be shown that the notion of a representationϕ : A → End(V ) of A is in fact equivalent to that of an A-module V by settinga ∗ v = ϕ(a)(v) for a ∈ A and v ∈ V . In the papers to follow, we will mostoften consider modules, but use the words “representations” and “modules”interchangeably.

A subspaceW of anA-moduleV is called a submodule if it is itself a moduleunder the restricted A-action. A non-zero module which has no submodulesexcept for 0 and itself is called simple. Simple modules are of particularinterests in representation theory because many classes of modules can bedescribed via some filtration of simple modules. A module which decomposesinto a direct sum of simple modules is called semisimple.

For further reading on the representation theory of associative algebras, see[1].

2.3 CategoriesCategory theory is a formal framework for studying structural relationshipsbetween algebraic structures (and more general mathematical constructions)of the same kind. Adding additional layers of abstraction, it can even be used tostudy relationships between seemingly entirely different kinds of constructions.A category C is defined as a collection of the following data.

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• A class of objects Ob(C). In this thesis it will typically be the case that eachobject is an algebraic structure.

• For each ordered pair of objects (X,Y ), a class Hom(X,Y ) of morphisms,said to go from X to Y . In this thesis, Hom(X,Y ) will typically be the setof homomorphism from X to Y , where X and Y are algebraic structures.

• For each triple of objects (X,Y, Z) a composition map

◦ = ◦X,Y,Z : Hom(Y, Z)× Hom(X,Y ) → Hom(X,Z)

satisfying associativity, i.e. that for any f ∈ Hom(X,Y ), g ∈ Hom(Y, Z)and h ∈ Hom(Z,W ), we have h ◦ (g ◦ f) = (h ◦ g) ◦ f .

• For every object X , an identity morphism 1X ∈ Hom(X,X) satisfying forany object Y and any morphisms f ∈ Hom(X,Y ) and g ∈ Hom(Y,X) thatf ◦ 1X = f and 1X ◦ g = g.

Example 2.3.1. Let the class of all complex vector spaces be the class ofobjects, and for vector spaces X and Y let Hom(X,Y ) be the set of linearmaps fromX to Y . Taking ◦ to be function composition and 1X be the identityfunction, we obtain a category.

Example 2.3.2. For an algebra A, we denote by A-Mod the category whereobjects areA-modules and morphisms areA-module homomorphisms. Again,◦ and 1X is function composition and the identity function respectively.

Similarly to how homomorphisms are structure-preserving maps betweenalgebraic structures, one may define functors, which are structure-preservingmaps between categories. Functors that admit inverses are, as is the casefor other algebraic structures, called isomorphisms. However, the weakerequivalences of categories are functors that are seen more often in practiceand capture the notion of “sameness” of categories that tends to be relevant incategory theory.

We will in this thesis in particular also consider linear categories, whichare categories where every class of morphisms is a vector space such thatcomposition of morphisms is bilinear.

A more detailed account of the fundamentals of category theory can befound in [6].

2.4 Quiver algebras and path categoriesA quiver is a kind of directed graph so prominent in algebra that it has earnedits own name. A quiver Q is a tuple Q = (Q0, Q1, s, t), where the constitutingdata are specified as follows.• Q0 is the set of vertices.• Q1 is the set of arrows.

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• s, t : Q1 → Q0 are the source and target maps respectively (we say thatα ∈ Q1 has source s(α) and target t(α)).

A quiver is typically visualized by viewing the vertices as points and the arrowsas directed edges pointing from their source to their target.

To each quiver we may associate its quiver algebra (or path algebra), whichis an associative algebra, CQ, defined as follows.• The underlying vector space ofCQ is spanned by paths of one of the following

forms.(i) Concatenations α1α2 . . . αn of arrows in Q1 such that the source of αi

is the target of αi+1 for i = 1, 2, . . . , n− 1.(ii) For each vertex v in Q0 a so called empty path εv.

• The multiplication ◦ is defined on the first kind of paths by

α1α2 . . . αn ◦ β1β2 . . . βm = α1α2 . . . αnβ1β2 . . . βm

if the source of αn equals the target of β1;

α1α2 . . . αn ◦ εv = α1α2 . . . αn

if αn has source v;

εv ◦ α1α2 . . . αn = α1α2 . . . αn

if α1 has target v;εv ◦ εv = εv,

and all other multiplications of paths result in 0. This is then extended to amultiplication on the entire space CQ by bilinearity.As usual in ring theory, we may consider an ideal I of CQ, and then form

the quotient CQ/I , which will again be an algebra. The following Theorem,due to Gabriel (see [5]), is arguably the main reason for defining quivers andstudying their algebras.

Theorem 2.4.1. LetA be a finite-dimensional unital associative algebra. Thenthere exists a quiver Q and an ideal I of CQ such that

A-Mod ∼= CQ/I-Mod.

In the situation of the above theorem, Q is called the Gabriel quiver of A,and I is said to contain the relations corresponding to that quiver. Finding(the infinite-dimensional case analogues of) Gabriel quivers with relations isa main objective in both papers I and III.

For references as well as further reading on the role of quivers in the repre-sentation theory of associative algebras, see [1].

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2.5 Groups and group algebrasThe group is one of the most classical types of algebraic structures, and isoften used to model the “symmetries” of some construction (this latter claimis particularly transparent for the dihedral groups described below).

A group G is a set endowed with a binary multiplication operation _ · _,with a distinguished element g−1 ∈ G for each g ∈ G and furthermore anotherdistinguished element e ∈ G, subject to the following axioms.(i) e · g = e = g · e,(ii) g · g−1 = e = g−1 · g,(iii) a · (b · c) = (a · b) · c, for all g, a, b, c ∈ G.

The complex representation theory of finite groups is a very classical subject,and this is the part of group representation theory that will be of concern inthis thesis. The natural notion of a linear representation of a group G turns outto be equivalent to a representation of a certain associative algebra, as it wasdefined in Section 2.1. This algebra C[G] is called the group algebra of G, andis defined to be the vector space spanned by the elements of G and with algebramultiplication given as the bilinear extension of the group multiplication. Thisis indeed one of several related reasons why associative algebras have a moredistinguished position in representation theory than other algebraic structures.One can prove that every finite-dimensional complex module over a finite groupdecomposes into a direct sum of modules which are simple, i.e. which haveno nontrivial submodules. The representation theory of the finite groups thuslargely boils down to the study of the simple modules.

Example 2.5.1. (The symmetric groups.) For any positive integer n, the set ofpermutations on n elements, i.e. bijections from the set {1, 2, . . . , n} to itself,forms a group, Sn, under the multiplication given by composition of functions.This group is called the symmetric group on n elements.

The simple modules over Sn are indexed by partitions of n, or equivalentlyby the Young diagrams of size n. These are typically drawn as top/left adjustedrows (of weakly decreasing length) of boxes. For examples of these drawings,see the vertices of the graph illustrated in Figure 2.8 of Section 2.8. In Paper III,where we study the representation theory of injections between finite sets, therepresentation theory of Sn finds applications in a natural way since bijectionsare a special case of injections.

Example 2.5.2. (The dihedral groups.) Let n ≥ 3 be an integer. The dihedralgroup D2n of order 2n is the group of symmetries of the regular n-gon in theplane. Alternatively, it may be defined by the presentation

D2n = 〈rn, sn|rnn = 1, snrnsn = r−1n 〉,

i.e. as the quotient of the free group generated by the symbols rn and sn by therelations rnn = 1 and snrnsn = r−1

n . Note that we consider dihedral groups

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for n = 1, 2 undefined, in contrast to going the Coxeter route where it is naturalto define dihedral groups also for these n. The dihedral groups are the mainobjects of study in Paper IV.

For a more detailed account of the fundamentals of representations of groupsin general and those of the symmetric group in particular, see [10].

2.6 Lie algebras and universal enveloping algebrasLie algebras arise as the “linearizations” of Lie groups, and have as such oftenquite direct applications to areas such as quantum physics. A Lie algebra g isa vector space with the Lie bracket binary operation, [_, _]. The Lie bracket isbilinear and furthermore satisfies the following additional axioms.(i) [a, b] = −[b, a],(ii) [a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0,

for all a, b, c ∈ g.It turns out that the natural notion of representation theory of Lie algebras

may be equivalently described in terms of representations of corresponding as-sociative universal enveloping algebras. Because of this, one typically studiesrepresentations of universal enveloping algebras in lieu of Lie algebra represen-tations. For a Lie algebra, g, let F (g) be the free associative algebra on a basisof g, and let I be the ideal generated by elements of the form ab− ba− [a, b]for a and b in the chosen basis of g. Then the universal enveloping algebra ofg is defined to be

U(g) = F (g)/I.

The best studied Lie algebras are the finite-dimensional semisimple ones,which may be characterized as finite-dimensional Lie algebras, g, such thatevery U(g)-module is a direct sum of simple modules. The finite-dimensionalsemisimple Lie algebras are classified in terms of the celebrated Dynkin dia-grams, and their representation theory serves as inspiration also for the repre-sentation theory of other Lie algebras, such as the non-semisimple Schrödingeralgebra.

Example 2.6.1. For a positive integer, n, let sln be the vector space consistingof all n×n-matrices with zero trace, furthermore endowed with the Lie algebrastructure given by the commutator

sln × sln → sln

(a, b) → [a, b] = a · b− b · a,

where · is ordinary matrix multiplication. This Lie algebra is arguably thequintessential example of a semisimple Lie algebra.

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For a positive integer n and g = sln, there is a vector space decomposition

U(g) ∼= U(n−)⊗ U(h)⊗ U(n+),

where n− ⊂ g is the Lie subalgebra consisting of the strictly lower triangularmatrices, h ⊂ g is the Lie subalgebra consisting of the zero trace diagonalmatrices, and n+ ⊂ g is the Lie subalgebra consisting of the strictly uppertriangular matrices. This so called triangular decomposition can be generalizedto a wide class of Lie algebras (see [9]), including all those that occur in thepresent thesis, and finds extensive use in the representation theory of suchalgebras.

Example 2.6.2. The (centrally extended) Schrödinger algebra s is the Liealgebra with basis

{e, h, f, p, q, z}, where [z, s] = 0 for all s ∈ s, and the rest of the Lie bracketis given by:

[h, e] = 2e, [e, f ] = h, [h, f ] = −2f,[e, q] = p, [e, p] = 0, [h, p] = p,[f, p] = q, [f, q] = 0, [h, q] = −q,

[p, q] = z.

This Lie algebra is not semisimple, and is the main object of study of Paper I.

For further details on the theory of Lie algebras and their universal envelop-ing algebras, see [7] and [4].

2.7 Homological algebra and KoszulitySince its origin in topology, the language of homological algebra has evolvedinto one of the main tools of abstract algebra. One may think of homologicalalgebra as a way of clearly expressing how a certain algebraic structure iscomposed of other (hopefully easier to understand) structures.

Example 2.7.1. (Short exact sequences.) Let A be an algebra (say finite-dimensional, unital and associative), and let L,M,N be A-modules withan injective homomorphism f : L ↪→ M and a surjective homomorphismg : M � N such that ker(g) = im(f). This data is typically called a shortexact sequence and written

0 → Lf−→ M

g−→ N → 0.

One can see that there is a vector space decomposition M ∼= L⊕N , and that(up to isomorphism) L ⊂ M and M/L = N .

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The sense in which the module M in Example 2.7.1 “is composed of”the modules L and N should now be clear. Similar, but more complicated,relationships between modules are captured by the notions of complexes andtheir homology. A (cochain) complex C• of A-modules Ci with i ∈ Z is acollection of homomorphisms di : Ci → Ci+1 such that di+1 ◦ di = 0 for alli ∈ Z. Such a complex is often written

. . .d−2−−→ C−1 d−1−−→ C0 d0−→ C1 d1−→ C2 d2−→ . . . .

The i:th homology1 of C• is defined to be

Hi(C•) = ker(di)/ im(di−1).

The short exact sequence of Example 2.7.1 may be viewed as a complex bysetting for instance C−1 = L, C0 = M , C1 = N , d−1 = f , d0 = g, and allother modules and maps set to zero. As such, the short exact sequence has i:thhomology zero for all i ∈ Z. In general, however, the homologies constituteadditional pieces of the “jigsaw puzzle of modules” that complexes describe.

Example 2.7.2. (Projective resolutions.) Projective resolutions are complexesof a particular importance, in that they describe modules that can be piecedtogether using projective modules2. A projective resolution of the A-moduleM is a complex P • such that the following hold.(i) H0(P •) ∼= M .(ii) Hi(P •) = 0 for all i < 0.(iii) P i = 0 for all i > 0.(iv) P i is projective for all i ≤ 0.

In order to define the Koszul property of unital associative algebras we needto know what Z≥0-graded algebras and their correspondingly graded modulesare. An algebra A is Z≥0-graded if its underlying vector space decomposes as

A =⊕

i∈Z≥0

Ai,

and Ai · Aj ⊂ Ai+j . A graded A-module, then, is an A-module M whoseunderlying vector space decomposes as

M =⊕i∈Z

Mi

and Ai ∗Mj ⊂ Mi+j . Note that M , in contrast to A, may contain elements ofnegative degree. A Z≥0-graded algebra A is Koszul if A0 is semisimple and

1Technically, the correct term is cohomology, whereas “homology” would be used for a complexin the other direction, but let us not be too concerned with this quite inconsequential distinction.2The reader not familiar with projective modules may think of them as certain generalizationsof free modules.

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furthermore has a projective resolution P • of graded modules such that P i isgenerated by P i

i for every i ∈ Z. Koszul algebras have been described as “asclose to semisimple as a Z-graded algebra can be”. The Koszul property playsa main role in Papers II and III, albeit in the slightly generalized context ofpath categories rather than unital associative algebras.

For a thorough treatment of the fundamentals of homological algebra, see[6], and for an introduction to the theory of Koszul rings in general, see [2].

2.8 Induction and restriction of representationsConsider two unital and associative algebras B ⊂ A with the same unit. Foran A-module V , there is a natural way of obtaining a B-module structure onthe vector space V : simply let every b ∈ B act like b ∈ A. This restrictionprocedure gives rise to a functor

Res : A-Mod → B-Mod.

It turns out that there is a “universal” way also of assigning to every B-moduleV an A-module. This induction procedure gives rise to a functor

Ind : B-Mod → A-Mod,V → A⊗B V,

which is left adjoint to the restriction functor.

Example 2.8.1. (The case for the symmetric group algebras, and the Younglattice.) Let n be a positive integer and consider the symmetric groups as inExample 2.5.1. Then we have a natural inclusion of group algebras C[Sn] ⊂C[Sn+1], which gives rise to a restriction

Res : C[Sn+1]-Mod → C[Sn]-Mod,

and a corresponding induction functor in the other direction. The so calledYoung lattice is the graph obtained by taking all Young diagrams as vertices,and using the branching rule to draw an edge between two Young diagrams ifthe simple module corresponding to one of them is a direct summand of theresult of applying an induction or a restriction functor to the simple modulecorresponding to the other. This way, the Young lattice arises as the main objectof study in paper III. The (truncated) Young lattice is illustrated in Figure 2.8.

Example 2.8.2. (The case for the dihedral group algebras.) For integers n ≥ 3,consider the dihedral groups Dn as in example 2.5.2. For any integer p, there

17

Figure 2.1. The Young lattice truncated after diagrams of size four.

is a natural inclusionD2n ↪→ D2pn

rn → rppn

sn → spn.

With the geometrical interpretation of dihedral groups as the groups of sym-metries of regular n-gons, the above inclusion corresponds to inscribing theregular n-gon into the regular pn-gon. This inclusion induces a correspondinginclusion of group algebras C[D2n] ⊂ C[D2pn]. The restriction and inductionfunctors obtained from these inclusions form the main objects of study of paperIV, and certain graphs with vertices indexed by simple modules similar to theYoung lattice of the symmetric groups play a prominent role in this paper.

An introduction to the induction and restriction procedure for general alge-bras can be found in [12], and a proof of the branching rule for the symmetricgroups can be found in [10].

2.9 Category OFor nearly all Lie algebras, it is an overwhelming task to classify its entirecategory of modules. As is often done in such situations in algebra, mathe-maticians restrict their efforts to certain more tractable subcategories which

18

nevertheless have interesting properties. In the case of semisimple Lie alge-bras, one classical such category is the category O due to Bernstein, Gelfandand Gelfand, see [3].

Let g be a Lie algebra with a triangular decomposition U(g) ∼= U(n−) ⊗U(h) ⊗ U(n+) (see Section 2.6). Let also h∗ be the dual vector space of h.Then the category O ⊂ U(g)-Mod associated to this triangular decompositionis the subcategory consisting of all U(g)-modules3M satisfying the followingconditions.(i) M is finitely generated.(ii) M decomposes into a sum of weight spaces, i.e. M ∼= ⊕

λ∈h∗Mλ suchthat for every x ∈ h ⊂ U(g) and v ∈ Mλ we have xv = λ(x)v.

(iii) For every v ∈ M , the vector space U(n+)v is finite-dimensional.Any category O is abelian and contains important classes of modules.

Example 2.9.1. (Verma modules.) One important class of modules in O is theVerma modules. Let λ ∈ h∗ and let V be a one-dimensionalU(h⊕n+)-moduleV on which U(n+) acts by 0 and each x ∈ U(h) acts by the scalar λ(x). Thenthe module

Δ(λ) = U(g)⊗U(h⊕n+) V

is a Verma module. Note that we in effect obtain the Verma modules byapplying an induction functor as in Section 2.8.

A category O may be defined for the non-semisimple Schrödinger algebra(see Example 2.6.2), and this category is studied in Paper I.

A modern treatment of the category O for semisimple Lie algebras can befound in [8].

3Consistent with our habit (and sometimes abuse of notation) of studying the representationtheory of g via that of U(g), we define O to be a category of U(g)-modules, even though theessentially equivalent definition using g is more common.

19

3. Summary of papers

3.1 Paper IIn this paper we examine the blocks of the category O of the Schrödingeralgebra, s, and prove some consequences of the results thus obtained. Considerthe standard triangular decomposition

s ∼= n+ ⊕ h⊕ n−

of the Schrödinger algebra. In order to state the main results, we need to list afew definitions.

• Let h∗ be the dual vector space of h, and for the basis {h, z} in h, let{h�, z�} be the dual basis.

• Let R = {±2h�,±h�} be the set of roots• For ξ ∈ h∗/ZR, let O[ξ] ⊂ O be the full subcategory consisting of

modules M such that there is a nonzero v ∈ M satisfying that for anyx ∈ h we have xv = λ(x)v for some λ ∈ ξ.

• For λ ∈ h∗, let O[ξ]λ be the Serre subcategory of O[ξ] generated byΔ(λ), i.e. which contains Δ(λ) and is closed under taking short exactsequences with at least one module of the sequence in the subcategory.

• Let c be the Casimir element of s, and for λ ∈ h∗, let ϑλ be the scalarwith which c acts on Δ(λ).

It is easily seen that we have the decomposition

O ∼=⊕

ξ∈h∗/ZRO[ξ].

This decomposition may be refined by further decomposing each O[ξ], in waysthat depend on ξ. In particular, it is crucial whether h acts by a (half-)integeror not, and whether z acts by zero or not. We have the following results, wherethe central charge refers to the value λ(z) for any λ ∈ ξ ⊂ h∗.

Proposition 3.1.1. Let ξ ∈ h∗/ZR be of nonzero central charge. Assume thatλ(h) �∈ 1

2Z for any λ ∈ ξ. Then the following hold.(i) The module Δ(λ) is simple for any λ ∈ ξ.(ii) We have the decomposition

O[ξ] ∼=⊕λ∈ξ

O[ξ]λ

20

(iii) The functor defined on objects as N → Nλ and on morphisms in theobvious way provides an equivalence between O[ξ]λ and the category offinite dimensional complex vector spaces, so that O[ξ]λ ∼= C-mod, withthe latter being the category of finite-dimensional vector spaces.

Proposition 3.1.2. Let ξ ∈ h∗/ZR be of nonzero central charge and assumethat λ(h) ∈ Z + 1

2 for any λ ∈ ξ. For i ∈ Z+ denote by λi the element in ξ

such that λi(h) = −32 + i. Then we have the following:

(i) For λ ∈ ξ the module Δ(λ) is simple if and only if λ(h) ≤ −32 .

(ii) For each i ∈ N we have a non-split short exact sequence

0 → Δ(−λi − 3h�) → Δ(λi) → L(λi) → 0.

(iii) We have the decomposition

O[ξ] ∼=⊕i∈Z+

O[ξ]λi.

(iv) We haveO[ξ]λ0∼= C-mod, more precisely, the functor defined on objects

as N → Nλ and on morphisms in the obvious way provides an equiv-alence between O[ξ]λ0

and the category of finite dimensional complexvector spaces.

(v) For i ∈ N the category O[ξ]λiis equivalent to the category of finite

dimensional representations overC of the following quiver with relations:

•a

�� •b

�� ab = 0.

Proposition 3.1.3. Let ξ ∈ h∗/ZR be of nonzero central charge and assumethat λ(h) ∈ Z for any λ ∈ ξ. For i ∈ Z+ denote by λi the element in ξ suchthat λi(h) = −1 + i. Then we have the following:(i) The module Δ(λ) is simple for each λ ∈ ξ.(ii) We have the decomposition

O[ξ] ∼=⊕i∈Z+

O[ξ]λi.

(iii) We haveO[ξ]λi∼= C⊕C-mod for all i ∈ Z+, whereC⊕C-mod denotes

the category of finite-dimensional C⊕ C-modules.

For the next results, consider the following two quivers:

∞Q∞ : . . .a

�� -1a

��

b

�� 0a

��

b

�� 1b

��a

��. . .b

��

21

and

Q∞ : 0a

�� 1b

��a

�� 2b

��a

��. . .b

��

with imposed commutativity relation ab = ba (which includes the relationab = 0 for the vertex 0 in Q∞).

We denote by∞Q+∞-lfmod the category of locally finite dimensional∞Q∞-

modules (in which ab = ba) that are bounded from the right, that is modules inwhich i is represented by the zero vector space for all i � 0. We also denoteby Q∞-fmod the category of finite dimensional Q∞-modules (in which againab = ba), that is modules in which each i is represented by a finite dimensionalvector space and these vector spaces are zero for all but finitely many i.

Theorem 3.1.1. Let ξ ∈ h∗/ZR be of zero central charge and assumeλ(h) �∈ Z

for any λ ∈ ξ. Then the category O[ξ] is equivalent to ∞Q+∞-lfmod.

Denote by Of the full subcategory of O consisting of all finite-dimensionalmodules in O.

Theorem 3.1.2. The categories Of and Q∞-fmod are equivalent.

Consider next the following quiver which we call Γ:

0a

��

s

��

1a

��

b

��

s

��

2a

��

b

��

s

��

. . .b

��

-1b′

�� -2a′

��

t

��

b�� -3

a

��b

��

t

��

-4a

��b

��

t

��

. . .a

��

Proposition 3.1.4. Let ξ ∈ h∗/ZR be of zero central charge and assumeλ(h) ∈ Z for any λ ∈ ξ. The quiver Γ is the Gabriel quiver for the categoryO[ξ].

As a consequence of the above results, one may arrive at a description ofthe center, Z(s), of the Schrödinger algebra.

Theorem 3.1.3. We have Z(s) = C[z, c].

Let mλ be the ideal of Z(s) that is generated by z − λ(z) and c− ϑλ.

Theorem 3.1.4. The annihilator in U(s) of Δ(λ) is the ideal U(s)mλ.

22

Finally one may prove.

Theorem 3.1.5. Primitive ideals in U(s) with nonzero central charge areexactly the annihilators of simple highest weight modules with nonzero centralcharge.

Going into the proofs of these results is outside of the scope of this summary.

3.2 Paper IILet k be an algebraically closed field, let G be some simple connected graphwhich is orientable so that it contains an infinite directed walk, and constructa k-linear category as follows: First form a quiver by replacing the edges ofG with one arrow in each direction. Then consider the k-linear category C′generated by the path algebra of this quiver. Finally, obtain another k-linearcategory C by taking the quotient by the two following kinds of relations.

1. For arrows b : i → j and a : j → k such that i �= j, j �= k and k �= i

ib �� j

a �� k

set ab = 0.2. For arrows a1 : i → j, b1 : j → i, b2 : j → k and a2 : k → j such that

i �= j, j �= k and k �= i

ia1 �� jb1

b2

�� ka2

set a1b1 = a2b2.The following is the main result of this paper.

Theorem 3.2.1. The category C is Koszul.

To prove this result, we construct for each simple C-module a projectiveresolution that is linear up to arbitrary degree. The idea is to start with takingthe projective cover of a simple module, and considering the correspondingshort exact sequence as a (non-projective) resolution. We proceed to replacethe kernel in this sequence with a resolution (also in the form of a short exactsequence), and then keep doing the same thing with any non-projective modulethat appears. It turns out that with the proper choices, only three kinds of shortexact sequences appear in this way, and the end result is a resolution that is bothprojective and linear up to arbitrary degree. The idea is formalized by lettingthe replacement procedure be handled by mapping cones, and the iteration byinduction.

23

3.3 Paper IIIThe finite sets and injections between them form a category, which we denoteby I. LetQ be the quiver given by (the Hasse diagram of) the Young lattice, andlet C be the quotient of the path category of Q by the relations that compositionof two arrows corresponding to addition of two boxes to the same column iszero. The following theorem was recently shown to hold by Sam and Snowden,see [11].

Theorem 3.3.1. The Gabriel quiver of CI is Q, and CI is Morita equivalentto CC, i.e.

CI-Mod ∼= CC-Mod.

In this paper, we give a new and more direct proof of this theorem. This isdone by computing subquotients of indecomposable projective CI-modules,which is in turn done by translating the problem into the language of therepresentation theory of the symmetric group and then applying the Littlewood-Richardson rule.

Next, we study Koszulity properties of C. The following technical lemma iskey.

Lemma 3.3.1. Each arrow, say from λ to μ of the Young quiver Q may beassigned a “sign” sλμ = ±1 such that for any set {λ1, λ2, λ3, λ4} of Youngdiagrams such that λ2 and λ3 are obtained by adding a node to λ1, and λ4 isobtained by adding a node to λ2 and λ3, we have sλ2

λ4sλ1

λ2= −sλ3

λ4sλ1

λ3.

While C was previously known to be Koszul, we here construct explicitlinear resolutions of simple C-modules. We do this as follows. Let Pλ〈i〉denote the projective C-module generated at the object λ and in degree −i. Fixsome (Young) diagram ξ. Let Ii be the set of diagrams that can be obtained byadding −i nodes, no two of which to the same row, to ξ. Define

Pi =

{⊕λ∈Ii Pλ〈i〉, for i ≤ 0

0, for i > 0.

Fix non-zero elements vλ,iλ ∈ Pλ〈i〉(λ). Consider a diagram μ. If μ is asubquotient of Pλ〈i〉, we have a uniquely determined x ∈ C(λ, μ). Otherwiseset x = 0. Finally define vλ,iμ = x · vλ,iλ , which is a basis element of thesubquotient μ in Pλ〈i〉, provided that μ is a subquotient of Pλ, and 0 otherwise.

Define the maps

πi,λ : Pi → Pi+1

vμ,iμ → sλμvλ,i+1μ ,

24

where the sign sλμ is the one defined in Lemma 3.3.1, and also

δi =

{⊕λ∈Ii+1

πi,λ, for i < 0

0, for i ≥ 0.

Let Lξ denote the simple C-module at ξ and concentrated in degree zero.

Theorem 3.3.2. The modules Pi and maps δi form a linear resolution, P•, ofLξ.

Along the way, we obtain the following theorem, where CC! denotes thequadratic dual of CC.

Theorem 3.3.3. There is an isomorphism CC! ∼= (CC)op.

3.4 Paper IVInspired by an influential similar construction for the symmetric groups, weconsider the dihedral groups D2n and construct complex algebras AP,M asfollows.

For P being any set of primes, define AP to be the free algebra generatedby the symbols Resp and Indp with p ∈ P . The complexified Grothendieckgroup

G = C⊗Z Groth[⊕n≥3

D2n-Mod]

of dihedral groups becomes an AP -module with action induced by the actionsof Resp and Indp on the Grothendieck group. For any submodule M ⊂ G,let AnnAP

(M) be the ideal of elements of AP that annihilate each element ofM, and finally let

AP,M = AP /AnnAP(M).

Our main results hold when no p ∈ P or n with a D2n-module in M iseven, and furthermore for every n either all D2n-modules lie in M or nonedoes. Fix some arbitrary p′ ∈ P and define the algebras

TP,M = AP,M/〈Resp′ Indp′ −1〉.In this setting, our results are summarized in the following theorems.

The first theorem gives a decomposition of AP,M into indecomposablealgebras.

Theorem 3.4.1. There is an isomorphism of algebras

AP,M ∼= TP,M ⊕ TP,M,

25

and TP,M is an indecomposable algebra. Furthermore, the center of AP,M isgenerated by 1 and Resp Indp, where p ∈ P is arbitrary.

The next theorems describe the algebra TP,M explicitly in terms of a basisand relations. We get two cases depending on whether the prime factors of theinvolved dihedral group orders lie in P or not.

Theorem 3.4.2. Assume that there is no D2n-module in M with all primefactors ofn belonging toP . ThenTP,M has a basis consisting of the monomialsof the forms

Indl1p1. . . Ind

l|P |p|P | Res

k1p1

. . .Resk|P |p|P |

with ki, li ∈ N, and relations generated by the ones of the following forms.(i) Resp Resq = Resq Resp.(ii) Indp Indq = Indq Indp.(iii) Indq Resp = Resp Indq,

for p �= q.(iv) Resp Indp = 1.

In order to state the next theorem, we need to define one of the key conceptsof the paper. We call a terminal subsequence (i.e. a right monomial factor) z′of z a nadir in z with respect to p if the number of Indp minus the number ofResp in z′ is minimal over all terminal subsequences of z. If z′ is a nadir in zwith respect to all p ∈ P simultaneously, then we call z′ a total nadir in z.

Theorem 3.4.3. Assume that there is some D2n-module in M with all primefactors ofn belonging toP . ThenTP,M has a basis consisting of the monomialsof the forms(i) Indl1p1

. . . Indl|P |p|P | Res

k1p1

. . .Resk|P |p|P |

with ki, li ∈ N,(ii) Indl1p1

Resk1p1

. . . Indl|P |p|P | Res

k|P |p|P |

with ki, li ∈ N such that ki �= 0 �= li for at least two i,(iii) Reskpi

Indlpj

with i �= j, and k, l ∈ Z>0,(iv) Respi (mod |P |)+1

IndlpiReskpi

Indpi (mod |P |)+1

with k, l ∈ Z>0,(v) Indlpj

ReskpjIndl1p1

. . . Indl|P |p|P |

with j ∈ {1, 2, . . . , |P |}, with k, l ∈ Z>0, and li ∈ N such that lj = 0but li �= 0 for at least one i,

(vi) Resk1p1

. . .Resk|P |p|P | Ind

lpjReskpj

with j ∈ {1, 2, . . . , |P |}, with k, l ∈ Z>0, and ki ∈ N such that kj = 0but ki �= 0 for at least one i,

and relations generated by the ones of the following forms.(i) Resp Resq = Resq Resp.

26

(ii) Indp Indq = Indq Indp.(iii) z1 = z1,

where z2 is the result of reordering the factors of z1 in a way such that therelative order of factorsResp and Indp for each fixed p ∈ P is unchanged,and where either both or none of z1 and z2 has a total nadir.

(iv) Resp Indp = 1.

Two features of the actions of Resp and Indp on M are heavily used in theproofs of our results. The first is that Resp acts locally nilpotently on M; moreconcretely, applying Resp to a D2n-module where n/p is not an integer ≥ 3yields zero. It is to capture the significance of this that we define the conceptof the nadir of a monomial in AP , which was used in Theorem 3.4.3. Thesecond feature is that the pattern with which Resp and Indp act on modulesis, in a certain sense, invariant under shifts of the order of the underlyingdihedral group, so that their actions on M are determined by their action on afinite-dimensional subspace of M.

The paper is concluded by some more modest results and conjectures formore general P and M.

27

4. Sammanfattning på svenska (Summary inSwedish)

Denna sammanfattning utgör en något komprimerad och till svenska över-satt version av föregående kapitel 1 och 3. Den första delen ger en kortpopulärvetenskaplig introduktion till abstrakt algebra och delområdet repre-sentationsteori. Den andra delen sammanfattar resultaten hos avhandlingensfyra ingående vetenskapliga artiklar.

4.1 Populärvetenskaplig introduktionFör de flesta av oss (och högst sannolikt för mänskligheten som sådan) börjaderesan in i matematikens värld med en samling konkreta föremål – låt säga enhandfull stenar – och operationen där specifika antal av dem adderas. I en första,trevande abstraktion insåg vi att addition av naturliga tal modellerar additionav antal föremål oberoende av de fysiska egenskaperna hos föremålen; endastantalen räknas. De naturliga talen utgör tillsammans med additionsoperationenett grundläggande exempel på en algebraisk struktur.

Vi började därefter studera ytterligare operationer (subtraktion, multiplika-tion och division) och ytterligare abstraktioner av den fysiska världen som vikan tillämpa dessa på (negativa tal, rationella tal, reella tal, komplexa tal ochsenare matriser av tal). Den tidiga algebran kretsade till stor del kring studierav ekvationer med detta begränsade antal operationer. Under de senaste tvåårhundradena så har utvecklingen såväl inom dessa studier som på fysikens om-råde motiverat helt nya abstraktioner av fysiska företeelser, och den modernaalgebran omfattar en uppsjö av algebraiska strukturer.

Inom delområdet representationsteori studerar algebraiker särskilda alge-braiska strukturer – så kallade representationer – som var och en införlivarstrukturella egenskaper hos en annan algebraisk struktur av intresse. Detta görsi regel antingen eftersom den senare algebraiska strukturen uppträder genomsina representationer i direkta tillämpningar, eller eftersom representationernadelar några av den senare algebraiska strukturens inressanta egenskaper sam-tidigt som de är lättare att studera. Den mest klassiska och välanvända sortensrepresentationer är den som utgörs av särskilda vektorrum av matriser förseddamed operationen matrismultiplikation (eller mer allmänt linjära transforma-tioner försedda med operationen funktionssammansättning). Denna sorts rep-resentationer kan användas för att studera många olika algebraiska strukturer,däribland kogeralgebror, grupper och Liealgebror, och representationerna kani samtliga av dessa fall betraktas som representationer av associativa algebror.

28

4.2 Sammanfattning av artiklarDenna avhandling innehåller fyra vetenskapliga artiklar där olika associativaalgebror och deras representationer studeras. Artikel I studerar representation-steorin för en Liealgebra, artiklarna II och III behandlar egenskapen koszulitethos två algebror, och artikel IV studerar en familj av algebror som uppstår urinduktion och restriktion av representationer av dihedralgrupperna.

4.2.1 Artikel II denna artikel studerar vi kategorin O för Schrödinger-algebran. Närmarebestämt delar vi upp kategorin som en summa av olika "block", och beskriverGabriel-kogren för blocken. För fallet med "nollskild central laddning" hittarvi även kogrens relationer. Också för den ändligtdimensionella delen av Ohittar vi såväl Gabriel-kogret som dess relationer.

Ovanstående resultat används sedan för att hitta centret till den universellaomslutande algebran till Schrödinger-algebran, annihilatorerna till Verma-modulerna, samt de primitiva ideal till den universella omslutande algebransom har triviellt snitt med Schrödinger-algebrans center.

4.2.2 Artikel III denna artikel konstruerar vi en familj, C, av vägkategorier som kan betraktassom lokalt kvadratiskt duala till preprojektiva algebror. Vi bevisar att dessakategorier har Koszul-egenskapen.

I beviset konstruerar vi resolutioner av de enkla C-modulerna som är projek-tiva och linjära till godtyckligt hög position. Detta görs genom att med hjälpav kon-konstruktionen sätta samman korta exakta följer, som kan väljas så attde alla återfinns i tre hanterbara familjer.

4.2.3 Artikel IIILåt I vara kategorin av ändliga mängder och injektionerna mellan dem. Låtäven Q vara kogret som ges av Young-gittret, och C vara vägkategorin av Qmodulo relationerna där de vägar som motsvarar addition av två lådor i sammakolumn i ett Young-diagram sätts till noll. I denna artikel hittar vi ett nyttoch mer direkt bevis för att Q är Gabriel-kogret till I, och att CI och CC ärMorita-ekvivalenta.

Vi hittar därefter ett nytt bevis för det kända faktum att C har Koszul-egenskapen genom att konstruera explicita linjära resolutioner till de enklaC-modulerna, och bevisar på vägen att C är kvadratiskt självdual.

29

4.2.4 Artikel IVI denna artikel studerar vi en familj av algebror AP,M som definieras medhjälp av aktionen av restriktions- och induktionsfunktorer på moduler över di-hedralgrupperna D2n och där P är en mängd primtal och M är (en delmodulav grothendieckgruppen av) en samling moduler över dihedralgrupperna. När-mare bestämt låter viAP,M vara den fria algebra som genereras av restriktions-och induktionsfunktorerna Resp och Indp där p ∈ P modulo annihilatorn vidden naturliga aktion av dessa på M.

Våra huvudsakliga resultat gäller givet att inget p ∈ P eller n med någonD2n-modul i M är jämnt, samt att det för varje n gäller att antingen ingen ellersamtliga D2n-moduler finns i M. Dessa resultat inkluderar en uppdelning avAP,M i odelbara delalgebror, en explicit beskrivning av en bas och relationerför algebran, samt en beskrivning av dess center.

30

5. Acknowledgements

It is with great delight that I’m presented this opportunity to express myappreciation for some of the people who are the most important in my life,or to whom I’m otherwise indebted. You all deserve to hear this more oftenreally.

Walter, from the first course (in combinatorics!) where you taught me, Irecognized your extraordinary ability as a teacher. Whatever was your wayof thinking about mathematics, I knew I wanted to partake in it. You havesince impressed and inspired me also with you energy, your patience, and yourgenerosity, and have indeed shaped my way of thinking. Thank you for beingthe best advisor one could wish for.

Thank you also to my coadvisor, Martin, who has always been eager tohelp, and whose frequent interjections during seminars have reliably taught memore than the talks themselves. Thank you to my other colleagues at the mathdepartment for all the help and the relaxed and positive environment, not leastto the administration staff whose heroics keep the place running. A specialthanks to Jakob, Sam, Andrea, Erik, Marta, Anya, Sebastian, and the rest ofmy PhD student colleagues. I regret that I haven’t been in a position to fullyappreciate the warm camaraderie offered by your company.

Lastly, a loving thank you to all my friends and family, especially to mymother Liselotte, my late father Frantisek, my sister Susanna, and my brotherDennis. All and any faith I have in myself, I owe to the faith that you have putin me.

31

References

[1] Ibrahim Assem, Daniel Simson, and Andrzej Skowronski. Elements of theRepresentation Theory of Associative Algebras: Volume 1: Techniques ofRepresentation Theory. Cambridge University Press, 2006.

[2] Alexander Beilinson, Viktor Ginzburg, and Wolfgang Soergel. Koszul dualitypatterns in representation theory. Journal of the American MathematicalSociety, 9(2):473–527, 1996.

[3] Joseph Bernstein, Izrail Gel’fand, and Sergei Gel’fand. A category ofg-modules. Functional Analysis and its Applications, 10(2):87–92, 1976.

[4] Jacques Dixmier. Enveloping Algebras. North-Holland Publishing Company,1977.

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