The

natural habitat of this lesson is a bit further down the course, but it

was called into existence by a comment/question by

Kea

I don’t yet quite see where the nc

manifolds are, but I guess that’s coming.

As

I’m enjoying telling about all sorts of sources of finite dimensional

representations of $SL_2(\mathbb{Z}) $ (and will carry on doing so for

some time), more people may begin to wonder where I’m heading. For this

reason I’ll do a couple of very elementary posts on simple examples of

noncommutative manifolds.

I realize it is ‘bon ton’ these days

to say that noncommutative manifolds are virtual objects associated to

noncommutative algebras and that the calculation of certain invariants

of these algebras gives insight into the topology and/or geometry of

these non-existent spaces. My own attitude to noncommutative geometry is

different : to me, noncommutative manifolds are genuine sets of points

equipped with a topology and other structures which I can use as a

mnemotechnic device to solve the problem of interest to me which is the

classification of all finite dimensional representations of a smooth

noncommutative algebra.

Hence, when I speak of the

‘noncommutative manifold of $SL_2(\mathbb{Z}) $’ Im after an object

containing enough information to allow me (at least in principle) to

classify the isomorphism classes of all finite dimensional

$SL_2(\mathbb{Z}) $-representations. The whole point of this course is

to show that such an object exists and that we can make explicit

calculations with it. But I’m running far ahead. Let us start with

an elementary question :

**Riemann surfaces are examples of
noncommutative manifolds, so what is the noncommutative picture of
them?**

I’ve browsed the Google-pictures a bit and a picture

coming close to my mental image of the noncommutative manifold of a

Riemann surface locally looks like the picture on the left. Here, the checkerboard-surface is part of the Riemann surface

and the extra structure consists in putting in each point of the Riemann

surface a sphere, reflecting the local structure of the Riemann surface

near the point. In fact, my picture is slightly different : I want to

draw a loop in each point of the Riemann surface, but Ill explain why

the two pictures are equivalent and why they present a solution to the

problem of classifying all finite dimensional representations of the

Riemann surface. After all why do we draw and study Riemann

surfaces? Because we are interested in the solutions to equations. For

example, the points of the _Kleinian quartic Riemann

surface_ give us all solutions tex \in

\mathbb{C}^3 $ to the equation $X^3Y+Y^3Z+Z^3X=0 $. If (a,b,c) is such

a solution, then so are all scalar multiples $(\lambda a,\lambda

b,\lambda c) $ so we may as well assume that the Z$coordinate is equal

to 1 and are then interested in finding the solutions tex \in

\mathbb{C}^2 $ to the equation $X^3Y+Y^3+X=0 $ which gives us an affine

patch of the Kleinian quartic (in fact, these solutions give us all

points except for two, corresponding to the _points at infinity_ needed

to make the picture compact so that we can hold it in our hand and look

at it from all sides. These points at infinity correspond to the trivial

solutions (1,0,0) and (0,1,0)).

What is the connection

between points on this Riemann surface and representations? Well, if

(a,b) is a solution to the equation $X^3Y+Y^3+X=0 $, then we have a

_one-dimensional representation_ of the affine _coordinate ring_

$\mathbb{C}[X,Y]/(X^3Y+Y^3+X) $, that is, an algebra

morphism

$\mathbb{C}[X,Y]/(X^3Y+Y^3+X) \rightarrow \mathbb{C} $

defined by sending X to a and Y to b.

Conversely, any such one-dimensonal representation gives us a solution

(look at the images of X and Y and these will be the coordinates of

a solution). Thus, commutative algebraic geometry of smooth

curves (that is Riemann surfaces if you look at the ‘real’ picture)

can be seen as the study of one-dimensional representations of their

smooth coordinate algebras. In other words, the classical Riemann

surface gives us already the classifcation of all one-dimensional

representations, so now we are after the ‘other ones’.

In

noncommtative algebra it is not natural to restrict attention to algebra

maps to $\mathbb{C} $, at least we would also like to include algebra

maps to $n \times n $ matrices $M_n(\mathbb{C}) $. An n-dimensional

representation of the coordinate algebra of the Klein quartic is an

algebra map

$\mathbb{C}[X,Y]/(X^3Y+Y^3+X) \rightarrow M_n(\mathbb{C}) $

That is, we want to find all pairs of $n \times n $ matrices A and B satisfying the following

matrix-identities

$A.B=B.A $ and $A^3.B+B^3+A=0_n $

The

first equation tells us that the two matrices must commute (because we

took commuting variables X and Y) and the second equation really is

a set of $n^2 $-equations in the matrix-entries of A and

B.

There is a sneaky way to get lots of such matrix-couples

from a given solution (A,B), namely by _simultaneous conjugation_.

That is, if $C \in GL_n(\mathbb{C}) $ is any invertible $n \times n $

matrix, then also the matrix-couple $~(C^{-1}.A.C,C^{-1}.B.C) $

satisfies all the required equations (write the equations out and notice

that middle terms of the form $C.C^{-1} $ cancel out and check that one

then obtains the matrix-identities

$C^{-1} A B C = C^{-1} BA C $ and $C^{-1}(A^3B+B^3+A)C = 0_n $

which are satisfied because

(A,B) was supposed to be a solution). We then say that these two

n-dimensional representations are _isomorphic_ and naturally we are

only interested in classifying the isomorphism classes of all

representations.

Using classical commutative algebra theory of

Dedekind domains (such as the coordinate ring $\mathbb{C}[X,Y]/(X^3Y+Y^3+X) $)

allows us to give a complete solution to this problem. It says that any

n-dimensional representation is determined up to isomorphism by the

following geometric/combinatorial data

- a finite set of points $P_1,P_2,\dots,P_k $ on the Riemann surface with $k \leq n $.
- a set of positive integers $a_1,a_2,\dots,a_k $ associated to these pointssatisfying $a_1+a_2+\dots_a_k=n $.
- for each $a_i $ a partition of $a_i $ (that is, a decreasing sequence of numbers with total sum

$a_i $).

To encode this classification I’ll use the mental

picture of associating to every point of the Klein quartic a small

loop. $\xymatrix{\vtx{}

\ar@(ul,ur)} $ Don\’t get over-exited about this

noncommutative manifold picture of the Klein quartic, I do not mean to

represent something like closed strings emanating from all points of the

Riemann surface or any other fanshi-wanshi interpretation. Just as

Feynman-diagrams allow the initiated to calculate probabilities of

certain interactions, the **noncommutative manifold** allows the

initiated to classify finite dimensional representations.

Our

mental picture of the noncommutative manifold of the Klein quartic, that

is : the points of the Klein quartic together with a loop in each point,

will tell the initiated quite a few things, such as : The fact

that there are no arrows between distict points, tells us that the

classification problem splits into local problems in a finite number of

points. Technically, this encodes the fact that there are no nontrivial

extensions between different simples in the commutative case. This will

drastically change if we enter the noncommutative world…

The fact that there is one loop in each point, tells us that

the local classification problem in that point is the same as that of

classifying nilpotent matrices upto conjugation (which, by the Jordan

normal form result, are classified by partitions) Moreover,

the fact that there is one loop in each point tells us that the local

structure of simple representations near that point (that is, the points

on the Kleinian quartic lying nearby) are classified as the simple

representations of the polynmial algebra $\mathbb{C}[x] $ (which are the

points on the complex plane, giving the picture

of the Riemann sphere in each point reflecting the local

neighborhood of the point on the Klein quartic)

In general, the

noncommutative manifold associated to a noncommutative smooth algebra

will be of a similar geometric/combinatorial nature. Typically, it will

consist of a geometric collection of points and arrows and loops between

these points. This data will then allow us to reduce the classification

problem to that of _quiver-representations_ and will allow us to give

local descriptions of our noncommutative manifolds. Next time,

I’ll give the details in the first noncommutative example : the

skew-group algebra of a finite group of automorphisms on a Riemann

surface (such as the simple group $PSL_2(\mathbb{F}_7) $ acting on the

Klein quartic). Already in this case, some new phenomena will

appear…

** ADDED : ** While writing this post

NetNewsWire informed me that over at Noncommutative Geometry they have a

post on a similar topic : What is a noncommutative space.