Can

it be that one forgets an entire proof because the result doesn’t seem

important or relevant at the time? It seems the only logical explanation

for what happened last week. Raf Bocklandt asked me whether a

classification was known of all group algebras **l G** which are

noncommutative manifolds (that is, which are formally smooth a la Kontsevich-Rosenberg or, equivalently, quasi-free

a la Cuntz-Quillen). I said I didn’t know the answer and that it looked

like a difficult problem but at the same time it was entirely clear to

me how to attack this problem, even which book I needed to have a look

at to get started. And, indeed, after a visit to the library borrowing

Warren Dicks

lecture notes in mathematics 790 “Groups, trees and projective

modules” and browsing through it for a few minutes I had the rough

outline of the classification. As the proof is basicly a two-liner I

might as well sketch it here.

If **l G** is quasi-free it

must be hereditary so the augmentation ideal must be a projective

module. But Martin Dunwoody proved that this is equivalent to

**G** being a group acting on a (usually infinite) tree with finite

group vertex-stabilizers all of its orders being invertible in the

basefield **l**. Hence, by Bass-Serre theory **G** is the

fundamental group of a graph of finite groups (all orders being units in

**l**) and using this structural result it is then not difficult to

show that the group algebra **l G** does indeed have the lifting

property for morphisms modulo nilpotent ideals and hence is

quasi-free.

If **l** has characteristic zero (hence the

extra order conditions are void) one can invoke a result of Karrass

saying that quasi-freeness of **l G** is equivalent to **G** being

*virtually free* (that is, **G** has a free subgroup of finite

index). There are many interesting examples of virtually free groups.

One source are the discrete subgroups commensurable with **SL(2,Z)**

(among which all groups appearing in monstrous moonshine), another

source comes from the classification of rank two vectorbundles over

projective smooth curves over finite fields (see the later chapters of

Serre’s Trees). So

one can use non-commutative geometry to study the finite dimensional

representations of virtually free groups generalizing the approach with

Jan Adriaenssens in Non-commutative covers and the modular group (btw.

Jan claims that a revision of this paper will be available soon).

In order to avoid that I forget all of this once again, I’ve

written over the last couple of days a short note explaining what I know

of representations of virtually free groups (or more generally of

*fundamental algebras* of finite graphs of separable

**l**-algebras). I may (or may not) post this note on the arXiv in

the coming weeks. But, if you have a reason to be interested in this,

send me an email and I’ll send you a sneak preview.

# Tag: representations

Tomorrow

I’ll start with the course *Projects in non-commutative geometry*

in our masterclass. The idea of this course (and its companion

*Projects in non-commutative algebra* run by Fred Van Oystaeyen) is

that students should make a small (original if possible) work, that may

eventually lead to a publication.

At this moment the students

have seen the following : definition and examples of quasi-free algebras

(aka formally smooth algebras, non-commutative curves), their

representation varieties, their connected component semigroup and the

Euler-form on it. Last week, Markus Reineke used all this in his mini-course

Rational points of varieties associated to quasi-free

algebras. In it, Markus gave a method to compute (at least in

principle) the number of points of the *non-commutative Hilbert
scheme* and the varieties of

*simple representations*over a

finite field. Here, in principle means that Markus demands a lot of

knowledge in advance : the number of points of all connected components

of all representation schemes of the algebra as well as of its scalar

extensions over finite field extensions, together with the action of the

Galois group on them … Sadly, I do not know too many examples were all

this information is known (apart from path algebras of quivers).

Therefore, it seems like a good idea to run through Markus’

calculations in some specific examples were I think one can get all this

:

*free products of semi-simple algebras*. The motivating examples

being the groupalgebra of the (projective)

*modular group*

**PSL(2,Z) = Z(2) * Z(3)**and the free matrix-products $M(n,F_q) *

M(m,F_q)$. I will explain how one begins to compute things in these

examples and will also explain how to get the One

quiver to rule them all in these cases. It would be interesting to

compare the calculations we will find with those corresponding to the

path algebra of this

*one quiver*.

As Markus set the good

example of writing out his notes and posting them, I will try to do the

same for my previous two sessions on quasi-free algebras over the next

couple of weeks.

Again I

spend the whole morning preparing my talks for tomorrow in the master

class. Here is an outline of what I will cover :

– examples of

noncommutative points and curves. Grothendieck’s characterization of

commutative regular algebras by the lifting property and a proof that

this lifting property in the category **alg** of all l-algebras is

equivalent to being a noncommutative curve (using the construction of a

generic square-zero extension).

– definition of the affine

scheme **rep(n,A)** of all n-dimensional representations (as always,

**l** is still arbitrary) and a proof that these schemes are smooth

using the universal property of **k(rep(n,A))** (via generic

matrices).

– whereas **rep(n,A)** is smooth it is in general

a disjoint union of its irreducible components and one can use the

sum-map to define a semigroup structure on these components when

**l** is algebraically closed. I’ll give some examples of this

semigroup and outline how the construction can be extended over

arbitrary basefields (via a cocommutative coalgebra).

–

definition of the Euler-form on **rep A**, all finite dimensional

representations. Outline of the main steps involved in showing that the

Euler-form defines a bilinear form on the connected component semigroup

when **l** is algebraically closed (using Jordan-Holder sequences and

upper-semicontinuity results).

After tomorrow’s

lectures I hope you are prepared for the mini-course by Markus Reineke on non-commutative Hilbert schemes

next week.

Never thought that I would ever consider Galois descent of *semigroup
coalgebras* but in preparing for my talks for the master-class it

came about naturally. Let

**A**be a formally smooth algebra

(sometimes called a quasi-free algebra, I prefer the terminology

noncommutative curve) over an arbitrary base-field

**k**. What, if

anything, can be said about the connected components of the affine

**k**-schemes

**rep(n,A)**of

**n**-dimensional representations

of

**A**? If

**k**is algebraically closed, then one can put a

commutative semigroup structure on the connected components induced by

the

*sum map*

rep(n,A) x rep(m,A) -> rep(n + m,A) : (M,N) -> M + N

as introduced and studied by Kent

Morrison a long while ago. So what would be a natural substitute for

this if **k** is arbitrary? Well, define **pi(n)** to be the

*maximal* unramified sub **k**-algebra of **k(rep(n,A))**,

the coordinate ring of **rep(n,A)**, then corresponding to the

sum-map above is a map

pi(n + m) -> pi(n) \\otimes pi(m)

and these maps define on the *graded
space*

Pi(A) = pi(0) + pi(1) + pi(2) + ...

the

structure of a graded commutative **k**-coalgebra with

comultiplication

pi(n) -> sum(a + b=n) pi(a) \\otimes pi(b)

The relevance of **Pi(A)** is that if we consider it

over the algebraic closure **K** of **k** we get the *semigroup
coalgebra*

K G with g -> sum(h.h\' = g) h \\otimes h\'

where **G** is Morrison\’s connected component

semigroup. That is, **Pi(A)** is a **k**-form of this semigroup

coalgebra. Perhaps it is a good project for one of the students to work

this out in detail (and correct possible mistakes I made) and give some

concrete examples for formally smooth algebras **A**. If you know of

a reference on this, please let me know.

Yesterday I made reservations for lecture rooms to run the

master class on non-commutative geometry sponsored by the ESF-NOG project. We have a lecture room on

monday- and wednesday afternoon and friday the whole day which should be

enough. I will run two courses in the program : *non-commutative
geometry* and

*projects in non-commutative geometry*both 30

hours. I hope that Raf Bocklandt will do most of the work on the

*Geometric invariant theory*course so that my contribution to it

can be minimal. Here are the first ideas of topics I want to cover in my

courses. As always, all suggestions are wellcome (just add a

*comment*).

**non-commutative geometry** : As

I am running this course jointly with Markus Reineke and as Markus will give a

mini-course on his work on non-commutative Hilbert schemes, I will explain

the theory of *formally smooth algebras*. I will cover most of the

paper by Joachim Cuntz and Daniel Quillen “Algebra extensions and

nonsingularity”, Journal of AMS, v.8, no. 2, 1995, 251?289. Further,

I’ll do the first section of the paper by Alexander Rosenberg and Maxim Kontsevich,

“Noncommutative smooth spaces“. Then, I will

explain some of my own work including the “One

quiver to rule them all” paper and my recent attempts to classify

all formally smooth algebras up to non-commutative birational

equivalence. When dealing with the last topic I will explain some of Aidan Schofield‘s paper

“Birational classification of moduli spaces of representations of quivers“.

**projects in
non-commutative geometry** : This is one of the two courses (the other

being “projects in non-commutative algebra” run by Fred Van Oystaeyen)

for which the students have to write a paper so I will take as the topic

of my talks the application of non-commutative geometry (in particular

the theory of orders in central simple algebras) to the resolution of

commutative singularities and ask the students to carry out the detailed

analysis for one of the following important classes of examples :

quantum groups at roots of unity, deformed preprojective algebras or

symplectic reflexion algebras. I will explain in much more detail three talks I gave on the subject last fall in

Luminy. But I will begin with more background material on central simple

algebras and orders.