Generic representation theory of quivers with relations

Abstract

Communicated by Kent R. Fuller Keywords: Representations of finite dimensional algebras Irreducible components of parameterizing varieties Generic properties of representations Path algebras modulo relationsThe irreducible components of varieties parametrizing the finite dimensional representations of a finite dimensional algebra Λ are explored, with regard to both their geometry and the structure of the modules they encode. Provided that the base field is algebraically closed and has infinite transcendence degree over its prime field, we establish the existence and uniqueness (not up to isomorphism, but in a strong sense to be specified) of a generic module for any 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 statementa priori almost obvious -lies in the description of such a module in a format accessible to representation-theoretic techniques. Our approach to generic modules over path algebras modulo relations, 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 in terms of quiver and relations. The sharpest specific results on all fronts are obtained for truncated path algebras, that is, for path algebras of quivers modulo ideals generated by all paths of a fixed length; this class of algebras extends the comparatively thoroughly studied hereditary case, displaying many novel features.

Effective dimension of finite semigroups

“…Quite a bit is known about the geometry of representation varieties for truncated path algebras, see [5], and perhaps this can be of use for solving Question 32.…”

Closures in varieties of representations and irreducible components

“…Next, we recall more discerning graphical invariants associated to a finite dimensional Λ-module, namely its hypergraphs; see [1,Definition 3.9].…”

“…While hypergraphs pin down families of modules, as opposed to individual isomorphism classes, they provide a useful tool for communicating, in a visually suggestive format, the generic structure of the modules in the components. For our diagrammatic representations of hypergraphs, we refer to [1], [7], and to the example below. This example will serve as a staple in the sequel.…”

“…Moreover, G is unique relative to this property, up to such a Morita self-equivalence. We refer to [1,Theorem 5.12], and to [1, Section 4] for a more general statement addressing arbitrary path algebras modulo relations.…”

Understanding Finite Dimensional Representations Generically

“…However, in extending Kac's idea beyond the hereditary case, one needs to take into account that Rep d (Λ) consists of a plethora of irreducible components in general, and that hardly any relevant condition imposed on the corresponding modules can be expected to hold across dense subsets of all components. Hence the quest now targets the generic representations in each of the individual components, leading to the following program: (1). Find the irreducible components of Rep d (Λ) in a representation-theoretic format, that is, in terms of module invariants which cut the components out of the parametrizing variety.…”