Generic representation theory of quivers with relations

被引:17
|
作者
Babson, E. [2 ]
Huisgen-Zimmermann, B. [1 ]
Thomas, R. [3 ]
机构
[1] Univ Calif Santa Barbara, Dept Math, Santa Barbara, CA 93106 USA
[2] Univ Calif Davis, Dept Math, Davis, CA 95616 USA
[3] Univ Washington, Dept Math, Seattle, WA 98195 USA
基金
美国国家科学基金会;
关键词
Representations of finite dimensional algebras; Irreducible components of parameterizing varieties; Generic properties of representations; Path algebras modulo relations; FINITE-DIMENSIONAL ALGEBRAS; UNISERIAL REPRESENTATIONS; VARIETIES; SUBREPRESENTATIONS; GEOMETRY; MODULES;
D O I
10.1016/j.jalgebra.2009.05.044
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
The irreducible components of varieties parametrizing the finite dimensional representations of a finite dimensional algebra Lambda 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 statement a 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. (C) 2009 Elsevier Inc. All rights reserved.
引用
收藏
页码:1877 / 1918
页数:42
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