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 $D = EndA(X)$ 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) $A \rightarrow Mn(D) = EndD(XD)$. 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 $A \rightarrow Mn(D),~A \rightarrow Mm(E)$ 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

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 $\wis{rep}~A$ of all finite dimensional A-representations but this time using filtrations with successive quotients being bricks rather than simples.

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groups, noncommutative, rationality, representations, simples, topology

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Posted in geometry
Written on Thu, 09 February 2006 at 9:13 am
Tags: groups, noncommutative, rationality, representations, simples, topology
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