Tame biserial algebras

Abstract

A similar result is suspected if we drop the property “representationfinite” in classes 1 and 2, and we will start an investigation of biserial algebras of infinite representation type. The key to the result for representation-finite algebras was to observe that each representation-finite biserial algebra is special [13] (see also the basic notations below), which means that it can be presented by a quiver with “nice” relations. Special algebras play an important role in the modular representation theory of finite groups. Namely, each representation-finite block of a group algebra and some of the tame blocks (they occur only in characteristic 2) are special [7, 10,4]. Moreover in the complex representation theory of the Lorentz group the so-called Harish-Chandra modules are defined over (tame) special algebras [6]. In this paper we show that /l(A)62 for any special algebra A and that special algebras of infinite representation type are tame. Our paper consists of four parts. First we prove that each special algebra is a factor of a special symmetric algebra (Theorem 1.5). For the latter the methods of Gelfand and Ponomarev used in the classification of the indecomposable Harish-Chandra modules of the Lorentz group apply and furnish a complete list of indecomposable finitely generated modules over special algebras (Proposition 2.3) presented in Section 2. From this list we 480 0021~8693/85 $3.00