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GENERIC REPRESENTATION THEORY OF QUIVERS WITH RELATIONS
 

Summary: GENERIC REPRESENTATION THEORY
OF QUIVERS WITH RELATIONS
E. Babson, B. Huisgen-Zimmermann, and R. Thomas
Abstract. The irreducible components of varieties parametrizing the finite dimensional rep-
resentations of a finite dimensional algebra are explored, in terms of both their geometry
and the structure of the modules they encode; as is to be expected, close connections be-
tween the two aspects surface. In particular, we establish the existence and uniqueness (not
up to isomorphism, but in a strong sense to be specified) of a generic module for a given
irreducible component C, that is, of a module which displays all categorically defined generic
properties of the modules parametrized by C; the crucial point of the existence statement
a priori almost obvious lies in the description of such a module in a format accessible to
representation-theoretic techniques. Our approach, by way of minimal projective resolutions,
is largely constructive. It is explicit for large classes of algebras of wild type. We follow with
an investigation of the properties of such generic modules with regard to quiver and relations
of . The sharpest specific results on all fronts are obtained for truncated path algebras, that
is, path algebras of quivers modulo ideals generated by all paths of a fixed length.
1. Introduction
Let be a basic finite dimensional algebra over an algebraically closed field K. Tame-
ness of the representation type of the only situation in which one can, at least in
principle, meaningfully classify all finite dimensional representations of is a borderline

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics