What is Non-commutative Geometry?

An invitation for undergraduate students.

Clarisson Rizzie CanluboUniversity of the Philippines-Diliman

crpcanlubo@math.upd.edu.ph

1 Introduction

Today, I am going to talk about what non-commutative geometry is. I will keep my talkinformal and vague. First and foremost, I want to say that non-commutative geometry, whichI will abbreviate as NCG, is a relatively young area of Mathematics, barely 40 years of age. Itis a robust generalization of classical geometry, and by classical geometry I mean any geometryyou have encountered before today including algebraic geometry and differential geometry.

To get a feel of what NCG is, we will answer several perturbative questions. They are asfollows:

(1) What are non-commutative spaces? These are the objects of interest of NCG so it isimportant to know what they are.

(2) What are they good for?

(3) How do we study them?

2 What are non-commutative spaces?

Straight answer: No one knows.

Detailed answer: No one has a definite answer. Is it bad that no one has a definite answer? No.This allows the theory to have great flexibility.

People working on NCG have preferences− the so-called models of NCG. Let me give you anidea what I mean by a model. Let us consider the models of a torus.

(1) For amateur mathematicians:

(2) For algebraic geometers: it is k× × k× where k× is the group of units of a field k.

(3) For Lie theorists: they are abelian subgroups of a Lie group.

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(4) For number theorists: They are quotients of R2 by a full-rank lattice.

(5) For topologists: they are compact, connected, orientable genus 2 surfaces.

In classical geometry, these are all different models of a torus. In NCG, these are all thesame. What then are these so-called models of NCG? I will mention three models but thesethree is far from a complete list of models. these are just the well-established ones.

(1) Grothendieck’s model : In this paradigm, non-commutative spaces are geometric toposes(singular: topos). The notion of a topos requires not only a graduate level knowledge ofmathematics but also a decent understanding of modern algebraic geometry. In short, Iwill not try to tell you what they are.

(2) Kontsevich’s model : In this school of thought, non-commutative spaces are a plethora ofstructures one built after another. Some people working in this area uses commutativedifferential graded algebras (CDGA) or differential graded categories (DG categories).Informally, in this picture we have

NCspace =

(classicalspace

)+

(1storderstuff

)+

(2ndorderstuff

)+ higher stuff

(3) Connes’ model : This is the most established among these three models. In this domain,non-commutative spaces are operator algebras.

You probably haven’t heard of what algebras, let alone what operator algebras. So let megive you a brief description of what they are. An algebra A over C is a vector space (i.e one thathas vector addition (+) and scalar multiplication (·) ) together with an additional operation,called vector multiplication (×) that satisfies a bunch of compatibility conditions with (+) and(·). We will not enumerate these conditions as one can see them in any standard book in algebra.Example of algebras are:

(1) C[x] is an algebra over C.

(2) C(R), the vector space of continuous complex-valued functions on R is an algebra, (vector)multiplication is pointwise multiplication of functions. Note the this is commutative, hence,we call C(R) a commutative algebra.

(3) Mn(C), the space of n× n complex matrices is an algebra over C whose (vector) multipli-cation is the usual multiplication of matrices.

So, what makes an algebra an operator algebra? In this case, there are additional structures,called the conjugation ∗ and norm ‖·‖. The conjugation ∗ is an operation which is an involution,i.e. (a∗)∗ = a for any a ∈ A. Meanwhile, the norm ‖·‖ is required to satisfy a bunch of properties,one of which is the following identity

‖a∗a‖ = ‖a‖2

for any a ∈ A. On top of having these two additional structures, we require A to be completewith respect to the norm ‖·‖, i.e. any Cauchy sequence in A with respect to ‖·‖ converges.

(1) C[x] is NOT an operator algebra. (why?)

(2) C(R) is also NOT an operator algebra. But its subalgebra C0(R), the subalgebra consistingof functions vanishing at infinity is an operator algebra.

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(3) Mn(C) is an operator algebra whose conjugation is given by Hermitian transpose of ma-trices and whose norm can be taken to be any of the so-called matrix norms.

There is a theorem saying that any operator algebra is a subalgebra of the B(H), whereB(H) is the algebra of bounded linear operators on a Hilbert space H.

3 The basis of Connes’ formulation of NCG.

A very prominent theorem in mathematics, the so-called Gelfand duality, which states that wehave a bijection

Appealing to the fact that the collection of commutative operator algebras is a subcollectionof the collection of (not necessarily commutative) operator algebras, we will call those thatcorrespond, via the extension of the Gelfand duality, to general operator algebras the non-commutative spaces. In other words,

This duality has other manifestations. For example, smooth structure of a differentiablemanifold is completely determined by its algebra of smooth functions. In other words, we have

where C∞(M) denotes the algebra of smooth functions on M . A discussion of this result canbe found in Nestruev [4]. In algebraic geometry, the Hilbert Nullstellensatz gives the followingduality.

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Here, O(X) denotes the ring of regular functions on the affine variety X. A proof of this canbe found in Hartshorne [2]. Finally, from the theory of Riemann surfaces, we have the followingduality.

where M(X) denotes the field of meromorphic functions on X. A proof of this result can befound in Farkas and Kra [1].

Project 1. Explore this duality between geometric structures and commutative algebras.

4 What are they good for?

Let us start with a motivation from physics. From a physicist’s point-of-view, modern physicsconstitutes two complimentary areas: the Theory of Relativity and Quantum Physics. ClassicalPhysics is just an approximation of these two. It is well-accepted that classical physics isdeterministic while quantum physics is not. This is a misconception. In my blog, I showed thatnon-determinacy already manifest in the Newtonian formalism of classical physics. See this link.

Project 2. Classical physics has three main formalism: Newtonian, Langrangian and Hamil-tonian formalisms. I discussed in the aforementioned blog that non-determinacy manifest itselfin Classical Physics via the Newtonian formalism. A physicist friend of mine mentioned that itis also true in the Langrangian formalism. A nice project would be to investigate whether suchphenomenon also happen in the Hamiltonian formalism.

This shows that ubiquity of the quantum theory. In the point-of-view just mentioned, relativisticphysics and quantum physics are in the same footing. This is not the case according to NCG.From the point-of-view of non-commutative geometry, quantum physics is much more generalthat relativistic physics. Moreover, what should be regarded as ”classical” includes classicalphysics and relativistic physics. Let us explain why this is the case.

In the classical domain, the dynamics happen in an ambient space. That is, there is a spaceof positions. This is oftentimes a manifold M with an additional structure. For example, it is asymplectic manifold when one deals with the Hamiltonian formalism. The scalar quantities ofthis system are either continuous, or once-differentiable, or twice-differentiable, or in the extremeend, smooth functions. In other words, scalar quantities are elements of Ck(M), k = 0, 1, ...,∞.Vector quantities such as momentum, acceleration or force are sections of vector bundles. Whenone deals with the relativistic case, there is still an ambient space. The only difference is thatthe manifold in this case is pointed, and the manifold already contains time-coordinates.

Meanwhile, in the quantum domain there is no ambient space to speak of. Instead, one hasa state space. The state space contains all possible ”state” of the system under consideration.

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For example, suppose the system under consideration is a particle moving on a line. A stateis a collection of data describing the position, momentum, energy, temperature, or any otherphysical attribute of that particle. If for example ”sadness” is a physical attribute of a particle,the amount of sadness that particle has on a particular instance is a state. In this domain, theposition x and momentum p are now of equal footing. They are operators on a Hilbert spaces(the state space). Using Max Born’s postulate of quantum mechanics, we have [x, p] = i~, where~ is the Planck’s constant and [x, p] denotes the commutator of x and p, i.e. [x, p] = xp − px.In particular, xp 6= px. This implies that the algebra generated by x and p is non-commutative!

Project 3. The relation [x, p] = i~ is also known as the canonical commutation relation. Thisis intimately related to a very nice group, called the Heisenberg group. It is the group consistingof 3× 3 complex matrices 1 a c

1 b1

where a, b, c ∈ C under matrix multiplication. A nice project would be to explore this relationship.

5 How do we study non-commutative spaces?

The question posted in this section is too general. Let us sparse this into several questions.

(1) How do we construct useful non-commutative spaces?

(2) What are their symmetries?

(3) How do we tell them apart?

(4) What about additional structures to non-commutative spaces?

Let us first tackle the first question. To fully understand the theory behind non-commutativegeometry, it is ultimately important to look at examples. This begs the question how one canconstruct useful examples to look at. One such method is provided by deformation quantization.

Deformation is an old yet not fully-understood topic in geometry. It goes back to the celebratedworks of Kodaira and Spencer. It is more or less well-understood for complex manifolds. Thebasic idea of deformation theory is to study properties of a geometric object by deforming it.

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The above picture illustrates how a circle deform into a line. This family of curves is parametrizedby the closed interval [0, 1]. For t = 0 we have the horizontal line while for t = 1 we have a

point. The curve over t where 0 < t < 1 is a circle centered att2 + 1

2talong the vertical line

whose radius is1− t2

2t.

Project 4. Explore the relationship of this family with the famous magnetic potentials.

Meanwhile, quantization deals with slightly perturbing the relations you are interested in.For example, consider quantities x and p which commutes, i.e. xp − px = 0. Then, one canquantize the system they describe by slightly perturbing the relation at hand. For example,xp − px = i~ where ~ is the Planck’s constant. The minuteness of ~ says that x and p almostcommute. Their commutator is sufficiently small (this is the reason why in the macro world, onehardly notices these quantum properties). Deformation quantization is then the joint processone gets when we perform this two together.

Project 5. Work out one example of deformation quantization. Believe or not, this is a niceundergraduate thesis.

Another good source of non-commutative spaces are badly-behaving dynamical systems. Con-sider a particle moving on a torus whose motion is described by the following diagram.

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When we identify opposite edges of this square to form a torus, the red path will trace a pathon the torus. This path is popular in math and has a really nice name. Do you know what itis?

The path the particle will traverse is called a trefoil knot. Note that the red lines in the squareabove have rational slope. In particular, their slope is 2/3. This is the reason why the particletraced a closed path. If the slope is irrational, the particle will just continue winding around thetorus without returning to its original position. This is an example of an ill-behaved dynamicalsystem. We will not prove here but this give rise to a non-commutative space Aθ given as follows

Aθ = C∗⟨U, V |U∗ = U−1, V ∗ = V −1, UV = e2πiθV U

⟩provided the slope θ is irrational. With this, we have the following.

Project 6. Discuss why the particle moving at an irrational angle θ gives the non-commutativespace Aθ.

Another method to produce non-commutative spaces is using the so-called universal con-structions. Sometimes you only care about a particular detail and nothing else. If this is thecase, universal constructions work best. It is simple yet provides all that you need. Let usillustrate how this works. It can be shown by a rigorous application of Fourier analysis that thealgebra of continuous functions on the torus has the following presentation

C(T) = C∗⟨U, V |U∗ = U−1, V ∗ = V −1, UV = V U

⟩.

Quantizing the last relation as UV = e2πiθV U gives the same non-commutative space Aθ (thesame non-commutative space we got earlier).

Project 7. It will be a nice undergraduate thesis to show that the algebra of continuous functionson the torus is the one given above. This requires extensive knowledge of Fourier analysis.

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6 What are their symmetries?

Symmetries of non-commutative spaces are often called quantum groups. They are indispensablefor contemporary mathematics. They are studied by a lot a mathematicians for purposes goingbeyond non-commutative geometry. We will list some of their aspect they inherit from theirclassical analogues− groups.

6.1 Representation theory

Representation theory is the study of ”shadows” of a group. Sometimes, having a concretepresentation of the group is not enough to study it. By their very nature, groups are abstractobjects− a set with an operation obeying certain rules. The goal of representation theory isto represent these abstract groups concretely. Hence the name. In particular, representationtheory turns an abstract group G into a (not exactly) subgroup1 of GLn(C), the group of n×ninvertible matrices (here, n is taken to be a general index, i.e. it need not be finite). Note thatmatrices are highly amenable to computational methods.

Representation theory goes beyond groups. Almost every decent algebraic structure hasa corresponding representation theory. Quantum groups are no exemption. In fact, more istrue. It’s representation theory looks strikingly similar to that of groups. It was first studiedby mathematical physicists dealing with integrable systems. These striking similarity is whatcompelled them to call these objects quantum groups.

Project 8. Explore representations of quantum groups and the so-called crystal bases.

Project 9. Study a particular integrable system (for example, Hamilton systems) and see howquantum groups pop up. Note that the study of general dynamical systems belong to the theoryof differential equations. However, if you do it under my supervision it would be differentialgeometric in nature.

Project 10. (For aspiring graduate students) Study Hitchin integrable systems.

6.2 Galois theory

I don’t think I have to say a word what Galois theory is. Nonetheless, let me say that it is abeautiful interplay between field and group theory with far-reaching applications to geometryand solving polynomials to name a few. Another fascinating aspect of quantum groups is thatGalois theory extends to them.

1Not exactly a subgroup but a homomorphic image. I can explain this better in person.

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Unfortunately, Hopf-Galois theory requires advanced knowledge of algebra I don’t think it issuitable for undergraduates. However, if one is interested to pursue this topic to their graduatestudies, a starting point could be the following project.

Project 11. Study non-commutative roots of polynomials.

For this project, a starting discussion can be found in page 739 of [3].

6.3 Lie theory

Lie theory is among the well-studied subject in math whose beauty has attracted a huge amountinterest from geometers, analysts, and algebraists alike. Spanning a century and a half ofmathematical activity, it remains to be one of the active areas of research. It is interesting in itsown right, even without associating it to quantum groups or non-commutative geometry. Butthis is really out of the question. Lie theory is so essential a large portion of NCG is devoted toit.

Lie theory is the interplay between Lie groups and Lie algebras. Lie groups, though properlyregarded as smooth groups, is originally conceptualized as continuous groups. Unfortunately,contemporary mathematics has evolve to have a prejudice that groups are discrete, more so,finite. On the other hand, Lie algebras can be thought of as an infinitesimal group. The relationbetween a Lie group and its Lie algebras is portrayed in the following figure.

Here, G is the Lie group and g is its Lie algebra. The point of tangency is at the identity elemente ∈ G. One of the many successes of the theory of quantum groups is to unify Lie groups and Liealgebras in one elegant umbrella. One of the beautiful things one learns in studying Lie groupsand Lie algebras is the fact that they behave similarly. Despite being completely different fromeach other, their resemblance goes indefinitely deep. The theory of quantum groups gives aunifying background for these two. Indeed, not only that groups give examples of quantumgroups, Lie algebras also do. With these, one can really see why Lie algebras are regarded asinfinitesimal groups. For an undergraduate project, I propose the following.

Project 12. Study the relation between Lie groups and Lie algebras in the basic case of matrixgroups. Discuss how such relation manifest itself to quantum groups.

In particular, it is interesting to see what the so-called exponential map in Lie theory turnsinto for quantum groups, see for example page 40 of [5]. As I mentioned earlier, Lie theory isinteresting by itself. One can opt to do an undergraduate project in this field not necessarilygearing towards NCG. For example, I propose the following studies.

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Project 13. Lie theory is often regarded as the Galois theory for differential equations. In thisproject, I porpose the student to study a particular differential equation and look at its symmetrygroup. Afterwards, explain the tight relationship with a very important theorem in classicalmechanics, the so-called Noether’s Theorem.

Project 14. Study Hopf fibration in relation to Lie groups.

There are a lot of interesting objects related to the study of Lie groups. An example of suchobjects are the so-called quivers.

Project 15. Quivers are directed multi-graphs. They are closely related to the representationtheory of Lie groups and Lie algebras. For this project, study the representations of quivers. Iftime permits, see how they are related to the representations of Lie groups and Lie algebras.

6.4 Harmonic analysis

Harmonic analysis is one of the important subject of modern mathematics. Apart from theelegance of the theory, they also make the abstract study of Lie (or topological) groups amenableto analytic methods. Harmonic analysis complements representation theory. A very importantinstance of harmonic analysis is Fourier analysis, which can be summarize as the harmonicanalysis on the group S1, the group of complex numbers of modulus 1. Quantum group is notfree of harmonic analytic methods. However, this is a highly advanced topic which I deem notsuitable for undergraduate students. The community of experts in this field is extremely smallI wouldn’t be surprised if not a lot of people haven’t heard of what a compact quantum groupis. I just mentioned this for the sake of completeness. However, for a motivated undergraduatestudent who hopes to do his/her graduates studies in this field, I would be very happy to guidehim/her through the basic concepts surrounding this topic.

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6.5 Fusion system

So far, we have mentioned classical stuffs that carry over to quantum groups. Let us mentionsomething that the theory of quantum groups has to offer to classical math. Quantum groupsmake the study of the so-called fusion systems easier. Fusion systems are categories that aregenerated by simple objects and the way these objects are combined are really simple. Fusionsystems makes the study of modular representations easier. Recall that a representation of agroup identifies it as a subgroup1 of GLn(C). Modular representations is the case when wereplace C by a field of positive characteristic p. Without fusion systems, these are challengingobjects to study since in particular, formulas one use in representation theory should be modifiedas to avoid division by the characteristic of the field in question. It might be inappropriate forundergraduate students to study fusion systems in its full generality for quantum groups. Withthis, I propose the student to do it first in the classical level.

Project 16. Study the application of fusion systems to modular representations of a finite group.

7 How do we tell non-commutative spaces apart?

This is a very important question not just for NCG but for geometry in general. Before answeringthis question, let me try to make this question even more understandable by looking at what itmeans for familiar things you have encountered.

For example, how do you tell real vector spaces V and W apart? You can simply look attheir dimension and it those numbers don’t agree you can be certain the two vector spacesare non-isomorphic. Conversely, having those numbers being equal implies that V and W areisomorphic vector spaces.

Another example of such question is as follows: how do you tell groups apart? More precisely,given group G and H, how can you tell for sure that they are different? This might sound like aneasy task but actually its not, especially if the groups are given by a presentation. Note that ifthe order of G is different from the order of H, there is no way G and H are isomorphic groups.However, if the orders are the same this does not mean that G ∼= H. For example, consider thenon-isomorphic groups Z/4 and the Klein 4-group V .

Let us look at another example. Let F and E be fields. If the characteristics of F and E aredifferent then for sure, they are different fields. The converse, just like the previous example, isnot true in general.

With these three examples, the dimension, order and characteristic are called invariants ofwhatever object they are attached to. The first one, dimension, is called a complete invariantsince it completely distinguishes the object to which it is attached to. In this case, these arevector spaces. The latter two, order and characteristic, are called incomplete invariants forobvious reasons. Note that the incompleteness of the order and characteristic makes the studyof groups and fields far more interesting than that of vector spaces. Let us mention moreinteresting examples of invariants.

Given a group G, the collection of (isomorphism classes) of representations of G forms asemi-ring. There is a natural way to make this into a ring, called the representation ring ofG and denoted by Rep(G). For example, the representation ring of the cyclic group Cn isZ[X]/(Xn − 1).

Another example of an invariant is the so-called de Rham cohomology. Given a manifoldM , the de Rham cohomology H∗dR(M) of M is (often a finite-dimensional) differential gradedvector space. What is it useful for? Remember Gauss’ Law, that in the three-dimensionalspace, if the surface integral across a closed surface is non-zero there must be a charged particle

1Not quite, as I mentioned in section 6.1.

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inside (the parity of which is determined by the parity of the value of the surface integral). To beprecise, the value of the surface integral is the net charge of the particles inside the surface underconsideration. de Rham cohomology is the mathematical abstraction/generalization1 of this veryimportant theorem to much general manifolds other than the three-dimensional euclidean spaceR3.

An invariant which is widely used in algebraic topology is the so-called fundamental group2

denoted as π1(X) of a topological space X. It is illustrated below.

Composition of loops is by concatenation.The latter three invariants mentioned are not complete (and we certainly don’t want them

to be, this incompleteness is what makes studying these things worthwhile). The former twoinvariants have well-established analogues in NCG while the latter is a work in progress. Inparticular, generalizing the fundamental group for non-commutative spaces has been the focusof my PhD research.

The analogues of the representation ring and the de Rham cohomology delve into the so-called bivariant K-theory and cyclic homology. Bivariant K-theory studies vector bundles overnon-commutative spaces. Although it goes beyond this, this is enough as a motivation. Whatis a vector bundle? A vector bundle over a topological space X is a family of vector spacescontinuously parametrized by X. This is visualized in the following figure.

1This answer is due to Qiaochu Yuan. I can explain this further in person. See for example [6].2The reason for the subscript 1 is because this is just the first in a series of groups called homotopy groups.

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Vector bundles are important since, as I already mentioned, vector quantities in classical me-chanics are sections of vector bundles. A section of a vector bundle is nothing but a way to(continuously) pick a vector out of the vector space lying over a point, per point of the manifold.Meanwhile, cyclic homology is the generalization of de Rham cohomology for non-commutativespaces. With this, a student may ask what could they possibly do as an undergraduate projectwith these? I propose the following studies.

Project 17. State and prove the Hochschild-Kostant-Rosenberg Theorem. This is perhaps, inmy opinion, the best way to enter non-commutative geometry.

Bivariant is way an advanced topic for undergraduate students. However, if one is determinedto study it, say for their graduate studies, I propose

Project 18. Compute some examples of representation rings of groups.

The rest of the presentation is just an exploration of the many possibilities one can go bydoing NCG. I don’t intend to discuss this further but students are welcome to ask questions inperson if they are interested with this topics.

8 References

[1] H. Farkas and I. Kra. Riemann surfaces. Secong edition, Graduate Text in Mathematics, 71,Springer-Verlag, 1992.

[2] R. Hartshorne. Algebraic Geometry. Graduate Text in Mathematics, Springer, 1977.

[3] P. Hajac (Editor). Lecture Notes on Noncommutative Geometry and Quantum Groups.https://www.mimuw.edu.pl/ pwit/toknotes/toknotes.pdf, 2004.

[4] J. Nestruev. Smooth Manifolds and Observables. Graduate Text in Mathematics, Springer-Verlag, 2003.

[5] S. Shnider and S. Sternberg. Quantum Groups: From Coalgebras to Drinfeld Algebras. In-ternational Press Inc., 1993.

[6] Plenty. de Rham cohomology.