modular – Page 23 – neverendingbooks

the one quiver for GL(2,Z)

Published August 26, 2004 by lieven

Before the vacation I finished a rewrite of the One quiver to rule them
all note. The main point of that note was to associate to any qurve
$A$ (formerly known as a quasi-free algebra in the terminology of
Cuntz-Quillen or a formally smooth algebra in the terminology of
Kontsevich-Rosenberg) a quiver $Q (A)$ and a dimension vector $α_{A}$
such that $A$ is etale isomorphic (in a yet to be defined
non-commutative etale toplogy) to a ring Morita equivalent to the path
algebra $l Q (A)$ where the Morita setting is determined by the dimension
vector $α_{A}$ . These “one-quiver settings” are easy to
work out for a group algebra $l G$ if $G$ is the amalgamated free product
of finite groups $G = H_{1} ★_{H} H_{2}$ .

Here is how to do
this : construct a bipartite quiver with the left vertices corresponding
to the irreducible representations of $H_{1}$ , say $S_{1}, . ., S_{k}$ of
dimensions $(d_{1}, . ., d_{k})$ and the right vertices corresponding to the
irreducible representations of $H_{2}$ , $T_{1}, . ., T_{l}$ of dimensions
$(e_{1}, . ., e_{l})$ . The number of arrows from the $i$ -th left vertex to
the $j$ -th right vertex is given by the dimension of $H o m_{H} (S_{i}, T_{j})$
This is the quiver I call the Zariski quiver for $G$ as the finite
dimensional $G$ -representations correspond to $θ$ -semistable
representations of this quiver for the stability structure $θ = (d_{1}, . ., d_{k}; - e_{1}, . ., - e_{l})$ . The one-quiver $Q (G)$ has vertices
corresponding to the minimal $θ$ -stable dimension vectors (say
$α, β, . .$ of the Zariski quiver and with the number of arrows
between two such vertices determined by $δ_{α β} - χ (α, β)$ where $χ$ is the Euler form of the Zariski
quiver. In the old note I've included the example of the projective
modular group $P S L_{2} (Z) = Z_{2} ★ Z_{3}$ (which can easily be
generalized to the modular group $S L_{2} (Z) = Z_{4} ★_{Z_{2}} Z_{6}$ )
which turns out to be the double of the extended Dynkin quiver
$\tilde{A_{5}}$ . In the rewrite I've also included an example of a
congruence subgroup $Γ_{0} (2) = Z_{4} ★_{Z_{2}}^{H N N}$ which is an
HNN-extension. These are somehow the classical examples of interesting
amalgamated (HNN) groups and one would like to have plenty of other
interesting examples. Yesterday I read a paper by Karen Vogtmann called

Automorphisms of free groups and outer space in which I encountered
an amalgamated product decomposition for $G L_{2} (Z) = D_{8} ★_{Z_{2} \times Z_{2}} (S_{3} \times Z_{2})$ where $D_{8}$ is the diheder group of 8
elements. When I got back from vacation I found a reference to this
result in my mail-box from Warren Dicks. Theorem 23.1, p. 82, in Heiner
Zieschang, Finite Groups of Mapping Classes of Surfaces, LNM 875,
Springer, Berlin, 1981.

I worked out the one-quiver and it has
the somewhat strange form depicted above. It is perfectly possible that
I made mistakes so if you find another result, please let me know.

added material (febr 2007) : mistakes were made and
the correct one quiver can be found elsewhere on this blog.

more noncommutative manifolds

Published June 9, 2004 by lieven

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.

capita selecta

Published April 22, 2004 by lieven

Rather than going to the NOG
III Workshop I think it is more fun to give a talk for the Capita
Selecta-course for 2nd year students on “Monstrous Moonshine”. If
I manage to explain to them at least something, I think I am in good
shape for next year\’s Baby Geometry (first year) course. Besides,
afterwards I may decide to give some details of Borcherds\’ solution next year in my 3rd year
Geometry-course…(but this may just be a little bit
over-optimistic).
Anyway, this is what I plan to do in my
lecture : explain both sides of the McKay-observation
that

196 884 = 196 883 + 1

that is, I\’ll give
the action of the modular group on the upper-half plane and prove that
its fundamental domain is just C using the modular j-function (left hand
side) and sketch the importance of the Monster group and its
representation theory (right hand side). Then I\’ll mention Ogg\’s
observation that the only subgroups Gamma(0,p)+ of SL(2,Z)
for which the fundamental domain has genus zero are the prime divisors
p of teh order of the Monster and I\’ll come to moonshine
conjecture of Conway and Norton (for those students who did hear my talk
on Antwerp sprouts, yes both Conway and Simon Norton (via his
SNORT-go) did appear there too…) and if time allows it, I\’ll sketch
the main idea of the proof. Fortunately, Richard Borcherds has written
some excellent expository papers I can use (see his papers-page and I also discovered a beautiful
moonshine-page by Helena Verrill which will make my job a lot
easier.
Btw. yesterday\’s Monster was taken from her other monster story…

Tag: modular

the one quiver for GL(2,Z)

more noncommutative manifolds

capita selecta