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.
Category: featured
After yesterday’s post I had to explain today what
point-modules and line-modules are and that one can really
describe them as points in a (commutative) variety. Seemingly, the
present focus on categorical methods scares possibly interested students
away and none of them seems to know that this non-commutative projective
algebraic geometry once dealt with very concrete examples.
Let
us fix the setting : A will be a quadratic algebra, that is, A is
a positively graded algebra, part of degree zero the basefield k,
generated by its homogeneous part A_1 of degree one (which we take to be
of k-dimension n 1) and with all defining relations quadratic in these
generators. Take m k-independent linear terms (that is, elements of A_1)
: l1,…,lm and consider the graded left A-module
L = A/(Al1 + ... + Alm)
Clearly, the Hilbert series of this
module (that is, the formal power series in t with coefficient of t^a
the k-dimension of the homogeneous part of L of degree a) starts off
with
Hilb(L,t) = 1 + (n+1-m) t + ...
and
we call L a linear d-dimensional module if the Hilbert series is
the power series expansion of
1/(1-t)^{d +1} = 1 + (d+1)t +(d
+1)(d +2)/2 t^2 ...
In particular, if d=0 (that is, m=n) then L
is said to be a point-module and if d=1 (that is, m=n-1) then L
is said to be a line-module. To a d-dimensional linear module L
one can associate a d-dimensional linear subspace of ordinary (that is,
commutative) projective n-space P^n. To do this, identify
P^n = P(A 1^*)
the projective space of the n 1 dimensional space of
linear functions on the homogeneous part of degree one. Then each of the
linear elements li determines a hyperplane V(li) in P^n and the
intersection of the m hyperplanes V(l1),…,V(lm) is the wanted
subspace. In particular, to a point-module corresponds a point in
P^n and to a line-module a line in P^n. So, where
is the non-commutativity of A hidden? Well, if P is a point-module
P = P0 + P1 + P2 +...
(with all components P_a one dimensional)
then the twisted module
P' = P1 + P2 + P3 + ...
is
again a point-module and the map P–>P’ defines an automorphism on the
point variety. In low dimensions, it is often possible to
reconstruct A from the point-variety and automorphism. In higher
dimensions, one has to consider also the higher dimensional linear
modules.
When I explained all this (far clumsier as it was a
long time since I worked with this) I was asked for an elementary text
on all this. ‘Why hasn’t anybody written a book on all this?’ Well,
Paul Smith wrote such a book so have a look at his
homepage. But then, it turned out that the version one can download from
one of his course pages is a more recent and a lot more
categorical version. There is no more sight of a useful book on
non-commutative projective spaces and their linear modules which might
give starting students an interesting way to learn some non-commutative
algebra and the beginnings of algebraic geometry (commutative and
non-commutative). So, hopefully Paul still has the old version around
and will make it available… The only webpage on this I could find in
short time are the slides of a talk by Michaela Vancliff.
Yesterday morning I thought that I could use some discussions I had a
week before with Markus Reineke to begin to make sense of one
sentence in Kontsevich’ Arbeitstagung talk Non-commutative smooth
spaces :
It seems plausible that Borcherds’ infinite rank
algebras with Monstrous symmetry can be realized inside Hall-Ringel
algebras for some small smooth noncommutative
spaces
However, as I’m running on a 68K RAM-memory, I
didn’t recall the fine details of all connections between the monster,
moonshine, vertex algebras and the like. Fortunately, there is the vast
amount of knowledge buried in the arXiv and a quick search on Borcherds gave me a
list of 17 papers. Among
these there are some delightful short (3 to 8 pages) expository papers
that gave me a quick recap on things I once must have read but forgot.
Moreover, Richard Borcherds has the gift of writing at the same time
readable and informative papers. If you want to get to the essence of
things in 15 minutes I can recommend What
is a vertex algebra? (“The answer to the question in the title is
that a vertex algebra is really a sort of commutative ring.”), What
is moonshine? (“At the time he discovered these relations, several
people thought it so unlikely that there could be a relation between the
monster and the elliptic modular function that they politely told McKay
that he was talking nonsense.”) and What
is the monster? (“3. It is the automorphism group of the monster
vertex algebra. (This is probably the best answer.)”). Borcherds
maintains also his homepage on which I found a few more (longer)
expository papers : Problems in moonshine and Automorphic forms and Lie algebras. After these
preliminaries it was time for the real goodies such as The
fake monster formal group, Quantum vertex algebras and the like.
After a day of enjoyable reading I think I’m again ‘a point’
wrt. vertex algebras. Unfortunately, I completely forgot what all this
could have to do with Kontsevich’ remark…