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 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
$\mathcal{O}^lw = \bigcup{w’} \mathbf{r}{w’}$ 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.

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Artin, arxiv, moduli, noncommutative, quivers, rationality, representations, simples, topology

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Posted in geometry
Written on Fri, 03 February 2006 at 4:20 pm
Tags: Artin, arxiv, moduli, noncommutative, quivers, rationality, representations, simples, topology
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