representations – Page 18 – neverendingbooks

why nag? (1)

Published March 26, 2006 by lieven

Let us
take a hopeless problem, motivate why something like non-commutative
algebraic geometry might help to solve it, and verify whether this
promise is kept.

Suppose we want to know all solutions in invertible
matrices to the braid relation (or Yang-Baxter equation)

$X Y X = Y X Y$

All such solutions (for varying size of matrices)
form an additive Abelian category $\mathbf{rep}~B_3$ , so a big step forward would be to know all its
simple solutions (that is, those whose matrices cannot be brought in
upper triangular block form). A literature check shows that even this
task is far too ambitious. The best result to date is the classification
due to Imre Tuba and
Hans Wenzl of simple solutions of which the matrix size is at most
5.

For fixed matrix size n, finding solutions in $\mathbf{rep}~B_3$ is the same as solving a system of $n^2$ cubic
polynomial relations in $2n^2$
unknowns, which quickly becomes a daunting task. Algebraic geometry
tells us that all solutions, say $\mathbf{rep}_n~B_3$ form an affine closed subvariety of $n^2$ -dimensional affine space. If we assume that $\mathbf{rep}_n~B_3$ is a smooth variety (that is, a manifold) and
if we know one solution explicitly, then we can use the tangent space in
this point to linearize the problem and to get at all solutions in a
neighborhood.

So, here is an idea : assume that $\mathbf{rep}~B_3$ itself would be a non-commutative manifold, then
we might linearize our problem by considering tangent spaces and obtain
new solutions out of already known ones. But, what is a non-commutative
manifold? Well, by the above we at least require that for all integers n
the commutative variety $\mathbf{rep}_n~B_3$ is a commutative manifold.

But, there
is still some redundancy in our problem : if $(X,Y)$ is a
solution, then so is any conjugated pair $(g^{-1}Xg,g^{-1}Yg)$ where $g \in GL_n$ is a basechange matrix. In categorical terms, we are only
interested in isomorphism classes of solutions. Again, if we fix the
size n of matrix-solutions, we consider the affine variety $\mathbf{rep}_n~B_3$ as a variety with a $GL_n$ -action
and we like to classify the orbits of simple solutions. If $\mathbf{rep}_n~B_3$ is a manifold then the theory of Luna slices
provides a method, both to linearize the problem as well as to reduce
its complexity. Instead of the tangent space we consider the normal
space N to the $GL_n$ -orbit
(in a suitable solution). On this affine space, the stabilizer subgroup
$GL(\alpha)$ acts and there is a natural one-to-one
correspondence between $GL_n$ -orbits
in $\mathbf{rep}_n~B_3$ and $GL(\alpha)$ -orbits in the normal space N (at least in a
neighborhood of the solution).

So, here is a refinement of the
idea : we would like to view $\mathbf{rep}~B_3$ as a non-commutative manifold with a group action
given by the notion of isomorphism. Then, in order to get new isoclasses
of solutions from a constructed one we want to reduce the size of our
problem by considering a linearization (the normal space to the orbit)
and on it an easier isomorphism problem.

However, we immediately
encounter a problem : calculating ranks of Jacobians we discover that
already $\mathbf{rep}_2~B_3$ is not a smooth variety so there is not a
chance in the world that $\mathbf{rep}~B_3$ might be a useful non-commutative manifold.
Still, if $(X,Y)$ is a
solution to the braid relation, then the matrix $(XYX)^2$
commutes with both X and Y.

If $(X,Y)$ is a
simple solution, this means that after performing a basechange, $C=(XYX)^2$ becomes a scalar matrix, say $\lambda^6 1_n$ . But then, $(X_1,Y_1) = (\lambda^{-1}X,\lambda^{-1}Y)$ is a solution to

$XYX = YXY , (XYX)^2 = 1$

and all such solutions form a
non-commutative closed subvariety, say $\mathbf{rep}~\Gamma$ of $\mathbf{rep}~B_3$ and if we know all (isomorphism classes of)
simple solutions in $\mathbf{rep}~\Gamma$ we have solved our problem as we just have to
bring in the additional scalar $\lambda \in \mathbb{C}^*$ .

Here we strike gold : $\mathbf{rep}~\Gamma$ is indeed a non-commutative manifold. This can
be seen by identifying $\Gamma$
with one of the most famous discrete infinite groups in mathematics :
the modular group $PSL_2(\mathbb{Z})$ . The modular group acts by Mobius
transformations on the upper half plane and this action can be used to
write $PSL_2(\mathbb{Z})$ as the free group product $\mathbb{Z}_2 \ast \mathbb{Z}_3$ . Finally, using
classical representation theory of finite groups it follows that indeed
all $\mathbf{rep}_n~\Gamma$ are commutative manifolds (possibly having
many connected components)! So, let us try to linearize this problem by
looking at its non-commutative tangent space, if we can figure out what
this might be.

Here is another idea (or rather a dogma) : in the
world of non-commutative manifolds, the role of affine spaces is played
by $\mathbf{rep}~Q$ the representations of finite quivers Q. A quiver
is just on oriented graph and a representation of it assigns to each
vertex a finite dimensional vector space and to each arrow a linear map
between the vertex-vector spaces. The notion of isomorphism in $\mathbf{rep}~Q$ is of course induced by base change actions in all
of these vertex-vector spaces. (to be continued)

noncommutative topology (4)

Published February 9, 2006 by lieven

For a
qurve (aka formally smooth algebra) A a *block* is a (possibly infinite
dimensional over the basefield) left A-module X such that its
endomorphism algebra $D = End_A(X)$ is a division algebra and X
(considered as a right D-module) is finite dimensional over D. If a
block X is finite dimensional over the basefield, we call it a *brick*
(aka a *Schur representation*). We want to endow the set of all blocks
with a topology and look at the induced topology on the subset of
bricks. It is an old result due to Claus Ringel
that there is a natural one-to-one correspondence between blocks of A
and algebra epimorphisms (in the categorical sense meaning that identify
equality of morphisms to another algebra) $A \rightarrow M_n(D) =
End_D(X_D)$. This result is important as it allows us to define a
partial order on teh set of all A-blocks via the notion of
*specialization*. If X and Y are two A-blocks with corresponding
epimorphisms $A \rightarrow M_n(D),~A \rightarrow M_m(E)$ we say that Y
is a specialization of X and we denote $X \leq Y$ provided there is an
epimorphism $A \rightarrow B$ making the diagram below commute

$\xymatrix{& M_n(D) \\\ A \ar[ru] \ar[r] \ar[rd] & B \ar[u]^i
\ar[d]^p \\\ & M_m(E)} $

where i is an inclusion and p is a
onto. This partial ordering was studied by Paul Cohn, George Bergman and
Aidan Schofield who use
the partial order to define the _closed subsets_ of blocks to be
those closed under specialization.

There are two important
constructions of A-blocks for a qurve A. One is Aidan’s construction of
a universal localization wrt. a *Sylvester rank function* (and which
should be of use in noncommutative rationality problems), the other
comes from invariant theory and is related to Markus Reineke’s monoid in
the special case when A is the path algebra of a quiver. Let X be a
GL(n)-closed irreducible subvariety of an irreducible component of
n-dimensional A-representations such that X contains a brick (and hence
a Zariski open subset of bricks), then taking PGL(n)-equivariant maps
from X to $M_n(\mathbb{C})$ determines a block (by inverting all central
elements). Now, take a *sensible* topology on the set of all A-bricks.
I would go for defining as the open wrt. a block X, the set of all
A-bricks which become simples after extending by the epimorphism
determined by a block Y such that $Y \leq X$. (note that this seems to
be different from the topology coming from the partial ordering…).
Still, wrt. this topology one can then again define a *noncommutative
topology* on the Abelian category $\mathbf{rep}~A$ of all finite
dimensional A-representations
but this time using filtrations with successive quotients being bricks
rather than simples.

noncommutative topology (3)

Published February 3, 2006 by lieven

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.

Tag: representations

why nag? (1)

noncommutative topology (4)

noncommutative topology (3)