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# Tag: Klein

The Klein Four Group is an a
capella group from the maths department of Northwestern. Below a link to
one of their songs (grabbed from P.P. Cook’s Tangent Space
).

Finite
Simple Group (of order two)

A Klein Four original by
Matt Salomone

The path of love is never
smooth
But mine’s continuous for you
You’re the upper bound in the chains of my heart
You’re my Axiom of Choice, you know it’s true
But lately our relation’s not so well-defined
And
I just can’t function without you
I’ll prove my
proposition and I’m sure you’ll find
We’re a
finite simple group of order two
I’m losing my
identity
I’m getting tensor every day
And
without loss of generality
I will assume that you feel the same
way
Since every time I see you, you just quotient out
The faithful image that I map into
But when we’re
one-to-one you’ll see what I’m about
‘Cause
we’re a finite simple group of order two
Our equivalence
was stable,
A principal love bundle sitting deep inside
But then you drove a wedge between our two-forms
Now
everything is so complexified
When we first met, we simply
connected
My heart was open but too dense
Our system
was already directed
To have a finite limit, in some sense

I’m living in the kernel of a rank-one map
From my
domain, its image looks so blue,
‘Cause all I see are
zeroes, it’s a cruel trap
But we’re a finite simple
group of order two
I’m not the smoothest operator in my
class,
But we’re a mirror pair, me and you,
So
let’s apply forgetful functors to the past
And be a
finite simple group, a finite simple group,
Let’s be a
finite simple group of order two
(Oughter: “Why not
three?”)
I’ve proved my proposition now, as you
can see,
So let’s both be associative and free
And by corollary, this shows you and I to be
Purely
inseparable. Q. E. D.

Now
that my non-geometry
post
is linked via the comments in this
string-coffee-table post
which in turn is available through a
trackback from the Kontsevich-Soibelman
paper
it is perhaps useful to add a few links.

The little
I’ve learned from reading about Connes-style non-commutative geometry is
this : if you have a situation where a discrete group is acting with a
bad orbit-space (for example, $GL_2(\mathbb{Z})$ acting on the whole
complex-plane, rather than just the upper half plane) you can associate
to this a $C^*$-algebra and study invariants of it and interprete them
as topological information about this bad orbit space. An intruiging
example is the one I mentioned and where the additional noncommutative
points (coming from the orbits on the real axis) seem to contain a lot
of modular information as clarified by work of Manin&Marcolli and
Zagier. Probably the best introduction into Connes-style
non-commutative geometry
from this perspective are the Lecture on
Arithmetic Noncommutative Geometry
by Matilde Marcolli. To
algebraists : this trick is very similar to looking at the
skew-group algebra $\mathbb{C}[x_1,\ldots,x_n] * G$ if
you want to study the _orbifold_ for a finite group action on affine
space. But as algebraist we have to stick to affine varieties and
polynomials so we can only deal with the case of a finite group,
analysts can be sloppier in their functions, so they can also do
something when the group is infinite.

By the way, the
skew-group algebra idea is also why non-commutative algebraic
geometry
enters string-theory via the link with orbifolds. The
easiest (and best understood) example is that of Kleinian singularities.
The best introduction to this idea is via the Representations
of quivers, preprojective algebras and deformations of quotient
singularities
notes by Bill Crawley-Boevey.

Artin-style non-commutative geometry aka
non-commutative projective geometry originated from the
work of Artin-Tate-Van den Bergh (in the west) and Odeskii-Feigin (in
the east) to understand Sklyanin algebras associated to elliptic curves
and automorphisms via ‘geometric’ objects such as point- (and
fat-point-) modules, line-modules and the like. An excellent survey
paper on low dimensional non-commutative projective geometry is Non-commutative curves and surfaces by Toby
Stafford and
Michel Van den Bergh
. The best introduction is the (also
neverending…) book-project Non-
commutative algebraic geometry
by Paul Smith who
maintains a
noncommutative geometry and algebra resource page
page (which is
also available from the header).

Non-geometry
started with the seminal paper ‘Algebra extensions and
nonsingularity’, J. Amer. Math. Soc. 8 (1995), 251-289 by Joachim
Cuntz
and Daniel Quillen but which is not available online. An
online introduction is Noncommutative smooth
spaces
by Kontsevich and Rosenberg. Surely, different people have
different motivations to study non-geometry. I assume Cuntz got
interested because inductive limits of separable algebras are quasi-free
(aka formally smooth aka qurves). Kontsevich and Soibelman want to study
morphisms and deformations of $A_{\infty}$-categories as they explain in
their recent
paper
. My own motivation to be interested in non-geometry is the
hope that in the next decades one will discover new exciting connections
between finite groups, algebraic curves and arithmetic groups (monstrous
moonshine
being the first, and still not entirely understood,
instance of this). Part of the problem is that these three topics seem
to be quite different, yet by taking group-algebras of finite or
arithmetic groups and coordinate rings of affine smooth curves they all
turn out to be quasi-free algebras, so perhaps non-geometry is the
unifying theory behind these seemingly unrelated topics.

If you go
to Oberwolfach and the weather
predictions are as good as last
weeks
, try to bring your mountain-bike along! Here is a nice
1hr30 to 2hrs tour : from the institute to Walke (height 300m), follow
the road north to Rankach and at the Romanes Hof turn left to Hackerhof.
Next, off-road along the Hacker lochweg until you hit the road
Haaghutte-Mooshutte at an height of 855m (this should be doable in under
one hour). A few metres further, you have a view at the highest
mountains in the vicinity of the Institute : the Grosser Hundskopf
(947m) and Kleiner Hundskopf (926m) as on the left. Then, descend along
the Kirchhofweg over Moosbauerhof all the way down to the
Dohlenbacherhof where you hit the main road which brings you back to the
institute going SW. Please take a pump and repair material along, I
had 2 flat tires in 4 days. If you happen to have a GPS, you can
download the gpx-file.
You can feed this to Tom Carden’s Google Map
GPX Viewer
and study it in more detail (I made one wrong turn in the
descent and also briefly lost GPS reception in the forest near the top
causing the top waypoint (the lower waypoint is the Institute)).

If you
were not present and still want to see some of the talks or if you are
just curious in the outcome of Paul’s
frantic typing on his PowerBook, he has put his (selection of
talks)-notes
online
. Perhaps I’ll write down some of my own recollections of this
meeting later.

Klein’s
quartic $X$is the smooth plane projective curve defined by
$x^3y+y^3z+z^3x=0$ and is one of the most remarkable mathematical
objects around. For example, it is a Hurwitz curve meaning that the
finite group of symmetries (when the genus is at least two this group
can have at most $84(g-1)$ elements) is as large as possible, which in
the case of the quartic is $168$ and the group itself is the unique
simple group of that order, $G = PSL_2(\mathbb{F}_7)$ also known as
Klein\’s group. John Baez has written a [beautiful page](http://math.ucr.edu/home/baez/klein.html) on the Klein quartic and
its symmetries. Another useful source of information is a paper by Noam
Elkies [The Klein quartic in number theory](www.msri.org/publications/books/Book35/files/elkies.pd).
The quotient map $X \rightarrow X/G \simeq \mathbb{P}^1$ has three
branch points of orders $2,3,7$ in the points on $\mathbb{P}^1$ with
coordinates $1728,0,\infty$. These points correspond to the three
non-free $G$-orbits consisting resp. of $84,56$ and $24$ points.
Now, remove from $X$ a couple of $G$-orbits to obtain an affine open
subset $Y$ such that $G$ acts on its cordinate ring $\mathbb{C}[Y]$ and
form the Klein stack (or hereditary order) $\mathbb{C}[Y] \bigstar G$,
the skew group algebra. In case the open subset $Y$ contains all
non-free orbits, the [one quiver](www.matrix.ua.ac.be/master/coursenotes/onequiver.pdf) of this
qurve has the following shape $\xymatrix{\vtx{} \ar@/^/[dd] \\ \\ \vtx{} \ar@/^/[uu]}$ $\xymatrix{& \vtx{} \ar[ddl] & \\ & & \\ \vtx{} \ar[rr] & & \vtx{} \ar[uul]}$ $\xymatrix{& & \vtx{} \ar[dll] & & \\ \vtx{} \ar[d] & & & & \vtx{} \ar[ull] \\ \vtx{} \ar[dr] & & & & \vtx{} \ar[u] \\ & \vtx{} \ar[rr] & & \vtx{} \ar[ur] &}$ Here, the three components correspond to the three
non-free orbits and the vertices correspond to the isoclasses of simple
$\mathbb{C}[Y] \bigstar G$ of dimension smaller than $168$. There are
two such of dimension $84$, three of dimension $56$ and seven of
dimension $24$ which I gave the non-imaginative names \’twins\’,
\’trinity\’ and \’the dwarfs\’. As we want to spice up later this
Klein stack to a larger group, we need to know the structure of these
exceptional simples as $G$-representations. Surely, someone must have
written a paper on the general problem of finding the $G$-structure of
simples of skew-group algebras $A \bigstar G$, so if you know a
reference please let me know. I used an old paper by Idun Reiten and
Christine Riedtmann to do this case (which is easier as the stabilizer
subgroups are cyclic and hence the induced representations of their
one-dimensionals correspond to the exceptional simples).

Here the
story of an idea to construct new examples of non-commutative compact
manifolds, the computational difficulties one runs into and, when they
are solved, the white noise one gets. But, perhaps, someone else can
spot a gem among all gibberish…
[Qurves](http://www.neverendingbooks.org/toolkit/pdffile.php?pdf=/TheLibrary/papers/qaq.pdf) (aka quasi-free algebras, aka formally smooth
algebras) are the \’affine\’ pieces of non-commutative manifolds. Basic
examples of qurves are : semi-simple algebras (e.g. group algebras of
finite groups), [path algebras of
quivers](http://www.lns.cornell.edu/spr/2001-06/msg0033251.html) and
coordinate rings of affine smooth curves. So, let us start with an
affine smooth curve $X$ and spice it up to get a very non-commutative
qurve. First, we bring in finite groups. Let $G$ be a finite group
acting on $X$, then we can form the skew-group algebra $A = \mathbfk[X] \bigstar G$. These are examples of prime Noetherian qurves (aka
hereditary orders). A more pompous way to phrase this is that these are
precisely the [one-dimensional smooth Deligne-Mumford
stacks](http://www.math.lsa.umich.edu/~danielch/paper/stacks.pdf).
As the 21-st century will turn out to be the time we discovered the
importance of non-Noetherian algebras, let us make a jump into the
wilderness and consider the amalgamated free algebra product $A = (\mathbf k[X] \bigstar G) \ast_{\mathbf k G} \mathbfk H$ where $G \subset H$ is an interesting extension of finite groups. Then, $A$ is
again a qurve on which $H$ acts in a way compatible with the $G$-action
on $X$ and $A$ is hugely non-commutative… A very basic example :
let $\mathbb{Z}/2\mathbb{Z}$ act on the affine line $\mathbfk[x]$ by
sending $x \mapsto -x$ and consider a finite [simple
group](http://mathworld.wolfram.com/SimpleGroup.html) $M$. As every
simple group has an involution, we have an embedding
$\mathbb{Z}/2\mathbb{Z} \subset M$ and can construct the qurve
$A=(\mathbfk[x] \bigstar \mathbb{Z}/2\mathbb{Z}) \ast_{\mathbfk \mathbb{Z}/2\mathbb{Z}} \mathbfk M$ on which the simple group $M$ acts
compatible with the involution on the affine line. To study the
corresponding non-commutative manifold, that is the Abelian category
$\mathbf{rep}~A$ of all finite dimensional representations of $A$ we have
to compute the [one quiver to rule them
all](http://www.matrix.ua.ac.be/master/coursenotes/onequiver.pdf) for
$A$. Because $A$ is a qurve, all its representation varieties
$\mathbf{rep}_n~A$ are smooth affine varieties, but they may have several
connected components. The direct sum of representations turns the set of
all these components into an Abelian semigroup and the vertices of the
\’one quiver\’ correspond to the generators of this semigroup whereas
the number of arrows between two such generators is given by the
dimension of $Ext^1_A(S_i,S_j)$ where $S_i,S_j$ are simple
$A$-representations lying in the respective components. All this
may seem hard to compute but it can be reduced to the study of another
quiver, the Zariski quiver associated to $A$ which is a bipartite quiver
with on the left the \’one quiver\’ for $\mathbfk[x] \bigstar \mathbb{Z}/2\mathbb{Z}$ which is just $\xymatrix{\vtx{} \ar@/^/[rr] & & \vtx{} \ar@/^/[ll]}$ (where the two vertices
correspond to the two simples of $\mathbb{Z}/2\mathbb{Z}$) and on the
right the \’one quiver\’ for $\mathbf k M$ (which just consists of as
many verticers as there are simple representations for $M$) and where
the number of arrows from a left- to a right-vertex is the number of
$\mathbb{Z}/2\mathbb{Z}$-morphisms between the respective simples. To
make matters even more concrete, let us consider the easiest example
when $M = A_5$ the alternating group on $5$ letters. The corresponding
Zariski quiver then turns out to be $\xymatrix{& & \vtx{1} \\\ \vtx{}\ar[urr] \ar@{=>}[rr] \ar@3[drr] \ar[ddrr] \ar[dddrr] \ar@/^/[dd] & & \vtx{4} \\\ & & \vtx{5} \\\ \vtx{} \ar@{=>}[uurr] \ar@{=>}[urr] \ar@{=>}[rr] \ar@{=>}[drr] \ar@/^/[uu] & & \vtx{3} \\\ & & \vtx{3}}$ The Euler-form of this quiver can then be used to
calculate the dimensions of the EXt-spaces giving the number of arrows
in the \’one quiver\’ for $A$. To find the vertices, that is, the
generators of the component semigroup we have to find the minimal
integral solutions to the pair of equations saying that the number of
simple $\mathbb{Z}/2\mathbb{Z}$ components based on the left-vertices is
equal to that one the right-vertices. In this case it is easy to see
that there are as many generators as simple $M$ representations. For
$A_5$ they correspond to the dimension vectors (for the Zariski quiver
having the first two components on the left) $\begin{cases} (1,2,0,0,0,0,1) \\ (1,2,0,0,0,1,0) \\ (3,2,0,0,1,0,0) \\ (2,2,0,1,0,0,0) \\ (1,0,1,0,0,0,0) \end{cases}$ We now have all
info to determine the \’one quiver\’ for $A$ and one would expect a nice
result. Instead one obtains a complete graph on all vertices with plenty
of arrows. More precisely one obtains as the one quiver for $A_5$
$\xymatrix{& & \vtx{} \ar@{=}[dll] \ar@{=}[dddl] \ar@{=}[dddr] \ar@{=}[drr] & & \\\ \vtx{} \ar@(ul,dl)|{4} \ar@{=}[rrrr]|{6} \ar@{=}[ddrrr]|{8} \ar@{=}[ddr]|{4} & & & & \vtx{} \ar@(ur,dr)|{8} \ar@{=}[ddlll]|{6} \ar@{=}[ddl]|{10} \\\ & & & & & \\\ & \vtx{} \ar@(dr,dl)|{4} \ar@{=}[rr]|{8} & & \vtx{} \ar@(dr,dl)|{11} & }$
with the number of arrows (in each direction) indicated. Not very
illuminating, I find. Still, as the one quiver is symmetric it follows
that all quotient varieties $\mathbf{iss}_n~A$ have a local Poisson
structure. Clearly, the above method can be generalized easily and all
examples I did compute so far have this \’nearly complete graph\’
feature. One might hope that if one would start with very special
curves and groups, one might obtain something more interesting. Another
time I\’ll tell what I got starting from Klein\’s quartic (on which the
simple group $PSL_2(\mathbb{F}_7)$ acts) when the situation was sexed-up
to the sporadic simple Mathieu group $M_{24}$ (of which
$PSL_2(\mathbb{F}_7)$ is a maximal subgroup).