The Structure of Biserial Algebras |
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Authors: | Vila-Freyer, Roberto Crawley-Boevey, William |
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Affiliation: | Mathematical Institute, Oxford University 2429 St. Giles, Oxford OX1 3LB |
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Abstract: | By an algebra we mean an associative k-algebra with identity,where k is an algebraically closed field. All algebras are assumedto be finite dimensional over k (except the path algebra kQ).An algebra is said to be biserial if every indecomposable projectiveleft or right -module P contains uniserial submodules U andV such that U+V=Rad(P) and U V is either zero or simple. (Recallthat a module is uniserial if it has a unique composition series,and the radical Rad(M) of a module M is the intersection ofits maximal submodules.) Biserial algebras arose as a naturalgeneralization of Nakayama's generalized uniserial algebras[2]. The condition first appeared in the work of Tachikawa [6,Proposition 2.7], and it was formalized by Fuller [1]. Examplesinclude blocks of group algebras with cyclic defect group; finitedimensional quotients of the algebras (1)(4) and (7)(9)in Ringel's list of tame local algebras [4]; the special biserialalgebras of [5, 8] and the regularly biserial algebras of [3].An algebra is basic if /Rad( ) is a product of copies of k.This paper contains a natural alternative characterization ofbasic biserial algebras, the concept of a bisected presentation.Using this characterization we can prove a number of resultsabout biserial algebras which were inaccessible before. In particularwe can describe basic biserial algebras by means of quiverswith relations. |
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