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).

# Tag: Klein

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).