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: Quillen

Bill

Schelter was a remarkable man. First, he was a top-class mathematician.

If you allow yourself to be impressed, read his proof of the

*Artin-Procesi* theorem. Bill was also among the first to take

*non-commutative geometry* seriously. Together with Mike Artin he

investigated a notion of non-commutative integral extensions and he was

the first to focuss attention to *formally smooth algebras* (a

suggestion later taken up by a.o. Cuntz-Quillen and Kontsevich) and a

relative version with respect to algebras satisfying all identities of

*n x n* matrices which (via work of Procesi) led to *smooth@n*

algebras. To youngsters, he is probably best know as the co-inventor of

*Artin-Schelter regular algebras*. I still vividly remember an

overly enthusiastic talk by him on the subject in Oberwolfach, sometime

in the late eighties. Secondly, Bill was a genuine *Lisp-guru* and

a strong proponent of *open source software*, see for example his

petition against software patents. He maintanind

his own version of Kyoto Common Lisp which developed into Gnu

Common Lisp. A quote on its history :

GCL is

the product of many hands over many years. The original effort was known

as the Kyoto Common Lisp system, written by Taiichi Yuasa and Masami

Hagiya in 1984. In 1987 new work was begun by William Schelter, and that

version of the system was called AKCL (Austin Kyoto Common Lisp). In

1994 AKCL was released as GCL (GNU Common Lisp) under the GNU public

library license. The primary purpose of GCL during that phase of it’s

existence was to support the Maxima computer algebra system, also

maintained by Dr. Schelter. It existed largely as a subproject of

Maxima.

Maxima started as Bill’s version of

*Macsyma* an MIT-based symbolic computation program to which he

added many routines, one of which was **Affine** a package that

allowed to do *Groebner-like* computations in non-commutative

algebras (implementing *Bergman’s diamond lemma*) and which he

needed to get a grip on *3-dimensional Artin-Schelter regular
algebras*. Michel and me convinced Fred to acquire funds to

buy us a work-station (costing at the time 20 to 30 iMacs) and have Bill

flown in from the States with his tape of

*maxima*and let him

*port*it to our

*Dec-station*. Antwerp was probably for years

the only place in the world (apart from MIT) where one could do

calculations in

*affine*(probably highly illegal at the time).

Still, lots of people benefitted from this, among others Michaela

Vancliff and Kristel Van Rompay in their investigation

of 4-dimensional Artin-Schelter regular algebras associated to an

automorphism of a quadric in three-dimensional projective space.

Yesterday I ran into Bill (alas virtually) by browsing the

*crypto*-category of

*Fink*. There it was, maxima, Bill’s package! I tried to install it

with the Fink Commander and failed but succeeded from the command line.

So, if you want to have your own version of it type

sudo fink install maxima

from the Terminal and it will install without

problems (giving you also a working copy of common lisp). Unfortunately

I do not remember too much of Macsyma or Affine but there is plenty of

documentation on the net. Manuals and user guides can be obtained from

the maxima homepage and the University of Texas

(Bill’s university) maintains an online manual, including a cryptic description of

some *Affine-commands*. But probably I’ll have to send Michaela an

email asking for some guidance on this… Here, as a tribute to Bill who

died in july 2001 the opening banner

iMacLieven:~ lieven$ /sw/bin/maxima Maxima 5.9.0 http://maxima.sourceforge.net Distributed under the GNU Public License. See the file COPYING. Dedicated to the memory of William Schelter. This is a development version of Maxima. The function bug_report() provides bug reporting information. (C1)Leave a Comment

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.