# Tag: simples

For
finite dimensional hereditary algebras, one can describe its
noncommutative topology (as developed in part 2)
explicitly, using results of Markus
Reineke
in The monoid
of families of quiver representations
. Consider a concrete example,
say

$A = \begin{bmatrix} \mathbb{C} & V \\ 0 & \mathbb{C} \end{bmatrix}$ where $V$ is an n-dimensional complex vectorspace, or
equivalently, A is the path algebra of the two point, n arrow quiver
$\xymatrix{\vtx{} \ar@/^/[r] \ar[r] \ar@/_/[r] & \vtx{}}$
Then, A has just 2 simple representations S and T (the vertex reps) of
dimension vectors s=(1,0) and t=(0,1). If w is a word in S and T we can
consider the set $\mathbf{r}_w$ of all A-representations having a
Jordan-Holder series with factors the terms in w (read from left to
right) so $\mathbf{r}_w \subset \mathbf{rep}_{(a,b)}~A$ when there are a
S-terms and b T-terms in w. Clearly all these subsets can be given the
structure of a monoid induced by concatenation of words, that is
$\mathbf{r}_w \star \mathbf{r}_{w’} = \mathbf{r}_{ww’}$ which is
Reineke’s *composition monoid*. In this case it is generated by
$\mathbf{r}_s$ and $\mathbf{r}_t$ and in the composition monoid the
following relations hold among these two generators
$\mathbf{r}_t^{\star n+1} \star \mathbf{r}_s = \mathbf{r}_t^{\star n} \star \mathbf{r}_s \star \mathbf{r}_t \quad \text{and} \quad \mathbf{r}_t \star \mathbf{r}_s^{\star n+1} = \mathbf{r}_s \star \mathbf{r}_t \star \mathbf{r}_s^{\star n}$ With these notations we can
now see that the left basic open set in the noncommutative topology
(associated to a noncommutative word w in S and T) is of the form
$\mathcal{O}^l_w = \bigcup_{w’} \mathbf{r}_{w’}$ where the union is
taken over all words w’ in S and T such that in the composition monoid
the relation holds $\mathbf{r}_{w’} = \mathbf{r}_w \star \mathbf{r}_{u}$
for another word u. Hence, each op these basic opens hits a large number
of $~\mathbf{rep}_{\alpha}$, in fact far too many for our purposes….
So, what do we want? We want to define a noncommutative notion of
birationality and clearly we want that if two algebras A and B are
birational that this is the same as saying that some open subsets of
their resp. $\mathbf{rep}$’s are homeomorphic. But, what do we
understand by *noncommutative birationality*? Clearly, if A and B are
prime Noethrian, this is clear. Both have a ring of fractions and we
demand them to be isomorphic (as in the commutative case). For this
special subclass the above noncommutative topology based on the Zariski
topology on the simples may be fine.

However, most qurves don’t have
a canonical ‘ring of fractions’. Usually they will have infinitely
many simple Artinian algebras which should be thought of as being
_a_ ring of fractions. For example, in the finite dimensional
example A above, if follows from Aidan Schofield‘s work Representations of rings over skew fields that
there is one such for every (a,b) with gcd(a,b)=1 and (a,b) satisfying
$a^2+b^2-n a b < 1$ (an indivisible Shur root for A).

And
what is the _noncommutative birationality result_ we are aiming
for in each of these cases? Well, the inspiration for this comes from
another result by Aidan (although it is not stated as such in the
paper…) Birational
classification of moduli spaces of representations of quivers
. In
this paper Aidan proves that if you take one of these indivisible Schur
roots (a,b) above, and if you look at $\alpha_n = n(a,b)$ that then the
moduli space of semi-stable quiver representations for this multiplied
dimension vector is birational to the quotient variety of
$1-(a^2+b^2-nab)$-tuples of $n \times n$-matrices under simultaneous
conjugation.

So, *morally speaking* this should be stated as the
fact that A is (along the ray determined by (a,b)) noncommutative
birational to the free algebra in $1-(a^2+b^2-nab)$ variables. And we
want a noncommutative topology on $\mathbf{rep}~A$ to encode all these
facts… As mentioned before, this can be done by replacing simples with
bricks (or if you want Schur representations) but that will have to wait
until next week.

A *qurve*
is an affine algebra such that $~\Omega^1~A$ is a projective
$~A~$-bimodule. Alternatively, it is an affine algebra allowing lifts of
algebra morphisms through nilpotent ideals and as such it is the ‘right’
noncommutative generalization of Grothendieck’s smoothness criterium.
Examples of qurves include : semi-simple algebras, coordinate rings of
affine smooth curves, hereditary orders over curves, group algebras of
virtually free groups, path algebras of quivers etc.

Hence, qurves
behave a lot like curves and as such one might hope to obtain one day a
‘birational’ classification of them, if we only knew what we mean
by this. Whereas the etale classification of them is understood (see for
example One quiver to
rule them all
or Qurves and quivers )
we don’t know what the Zariski topology of a qurve might be.

Usually, one assigns to a qurve $~A~$ the Abelian category of all its
finite dimensional representations $\mathbf{rep}~A$ and we would like to
equip this set with a topology of sorts. Because $~A~$ is a qurve, its
scheme of n-dimensional representations $\mathbf{rep}_n~A$ is a smooth
affine variety for each n, so clearly $\mathbf{rep}~A$ being the disjoint
union of these acquires a trivial but nice commutative topology.
However, we would like open sets to hit several of the components
$\mathbf{rep}_n~A$ thereby ‘connecting’ them to form a noncommutative
topological space associated to $~A~$.

In a noncommutative topology on
rep A
I proposed a way to do this and though the main idea remains a
good one, I’ll ammend the construction next time. Whereas we don’t know
of a topology on the whole of $\mathbf{rep}~A$, there is an obvious
ordinary topology on the subset $\mathbf{simp}~A$ of all simple finite
dimensional representations, namely the induced topology of the Zariski
topology on $~\mathbf{spec}~A$, the prime spectrum of twosided prime ideals
of $~A~$. As in commutative algebraic geometry the closed subsets of the
prime spectrum consist of all prime ideals containing a given twosided
ideal. A typical open subset of the induced topology on $\mathbf{simp}~A$
hits many of the components $\mathbf{rep}_n~A$, but how can we extend it to
a topology on the whole of the category $\mathbf{rep}~A$ ?

Every
finite dimensional representation has (usually several) Jordan-Holder
filtrations with simple successive quotients, so a natural idea is to
use these filtrations to extend the topology on the simples to all
representations by restricting the top (or bottom) of the Jordan-Holder
sequence. Let W be the set of all words w such as $U_1U_2 \ldots U_k$
where each $U_i$ is an open subset of $\mathbf{simp}~A$. We can now define
the *left basic open set* $\mathcal{O}_w^l$ consisting of all finite
dimensional representations M having a Jordan-Holder sequence such that
the i-th simple factor (counted from the bottom) belongs to $U_i$.
(Similarly, we can define a *right basic open set* by counting from the
top or a *symmetric basic open set* by merely requiring that the simples
appear in order in the sequence). One final technical (but important)
detail is that we should really consider equivalence classes of left
basic opens. If w and w’ are two words we will denote by $\mathbf{rep}(w \cup w’)$ the set of all finite dimensional representations having a
Jordan-Holder filtration with enough simple factors to have one for each
letter in w and w’. We then define $\mathcal{O}^l_w \equiv \mathcal{O}^l_{w’}$ iff $\mathcal{O}^l_w \cap \mathbf{rep}(w \cup w’) = \mathcal{O}^l_{w’} \cap \mathbf{rep}(w \cup w’)$. Equivalence classes of
these left basic opens form a partially ordered set (induced by
set-theoretic inclusion) with a unique minimal element 0 (the empty set
corresponding to the empty word) and a uunique maximal element 1 (the
set $\mathbf{rep}~A$ corresponding to the letter $w=\mathbf{simp}~A$).
Set-theoretic union induces an operation $\vee$ and the operation
$~\wedge$ is induced by concatenation of words, that is,
$\mathcal{O}^l_w \wedge \mathcal{O}^l_{w’} \equiv \mathcal{O}^l_{ww’}$.
This then defines a **left noncommutative topology** on $\mathbf{rep}~A$ in
the sense of Van Oystaeyen (see [part
1](http://www.neverendingbooks.org/index.php/noncommutative-topology-1 $). To be precise, it satisfies the axioms in the left and middle column of the following picture and similarly, the right basic opens give a right noncommutative topology (satisfying the axioms of the middle and right columns) whereas the symmetric opens satisfy all axioms giving the basis of a noncommutative topology. Even for very simple finite dimensional qurves such as$\begin{bmatrix} \mathbb{C} & \mathbb{C} \\ 0 & \mathbb{C}
\end{bmatrix}$this defines a properly noncommutative topology on the Abelian category of all finite dimensional representations which obviously respect isomorphisms so is really a noncommutative topology on the orbits. Still, while this may give a satisfactory local definition, in gluing qurves together one would like to relax simple representations to *Schurian* representations. This can be done but one has to replace the topology coming from the Zariski topology on the prime spectrum by the partial ordering on the *bricks* of the qurve, but that will have to wait until next time… Klein’s quartic$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)$is a maximal subgroup). I found an old copy (Vol 2 Number 4 1980) of the The Mathematical Intelligencer with on its front cover the list of the 26 _known_ sporadic groups together with a starred added in proof saying • added in proof … the classification of finite simple groups is complete. there are no other sporadic groups. (click on the left picture to see a larger scanned image). In it is a beautiful paper by John Conway “Monsters and moonshine” on the classification project. Along the way he describes the simplest non-trivial simple group$A_5 $as the icosahedral group. as well as other interpretations as Lie groups over finite fields. He also gives a nice introduction to representation theory and the properties of the character table allowing to reconstruct$A_5 \$ only knowing that there
must be a simple group of order 60.
A more technical account
of the classification project (sketching the main steps in precise
formulations) can be found online in the paper by Ron Solomon On finite simple
groups and their classification
. In addition to the posts by John Baez mentioned
in this
post
he has a few more columns on Platonic solids and their relation to Lie
algebras
, continued here.