The Representation Type of Group Algebras
Throughout this paper we use the term algebra to mean finite-dimensional algebra over a fixed algebraically closed field K and the term module to mean finitely generated right module. Algebras, as is usual in representation theory, are assumed to be basic
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,
ANDRZE~
SKOWRONSKI
NICHOLAS COPERNICUS UNIVERSITY 87~100
,
TORUN, POLAND
1. INTRODUCTION Throughout this paper we use the term algebra to mean finite-dimensional algebra over a fixed algebraically closed field K and the term module to mean finitely generated right module. Algebras, as is usual in representation theory, are assumed to be basic. For any algebra A we will denote by mod A the category of finitely generated les and by ind A the full subcategory of mod
A
A-modu~
consisting
of all indecomposable modules. Recall that an algebra A is called wild provided mod A has a full subcategory representation equivalent to the tegory of
finite-dimensia~a~ 1 as
ca~
K-vector spaces, modules
over the free associative algebra K(X,y) in two (non~commu~ ting)variables x andy. Further, A is of finite type if ind A has only finitely many nonisomorphic objects. If A
R. Göbel et al. (eds.), Abelian Groups and Modules © Springer-Verlag Wien 1984
518
A. Skowronski
is neither wild nor of finite type, then A is said to be tame. Drozd showed in [1,2] that an algebra A is tame if and only if A is not of finite type and for any dimension d, there exists a finite family of functors Fi: mod mod A, i
= 1, ••• ,nd,
where Ri
= K or
Ri~
Ri is a rational alge-
bra K[i]f of dimension 1, satisfying the conditions: (a) For each i, 1 ::l!t i ' nd, Fi
= ? ® Qi
where Qi is an Ri Ri-A-bimodule being a finitely generated free right Ri-module, (b) Every indecomposable A-module M of dimension d is of the form M ~ Fi(s) for some i and a simple Ri-module s. Let A be an algebra and let G be a finite group. We are concerned with the problem of determining the representation type of the group algebra AG. A characterization of group algebras AG of finite type has bean obtained in the case A
= K by
Higman [3] and in the general case by the
author and Meltzer (4,5]. Krugliak showed in [6] that if K is of charakteristic p) 2, then the group algebras KG of noncyclic p-groups are wild, and Brenner (7] has shown that for p
= 2,
the group algebras KG of all 2-groups beside the
cyclic, dihedral, semidihedral and quaternion groups are wild. Bondarenko [a] and Ringel [9) have indepadently shown that in characteristic 2 the group algebras of dihedral 2-groups are tame. Finally, Drozd and Bondarenko [10] have also shown that the group algebras of semidihedral and
Group Algebras
519
and quaternion 2-groups are tame in characteristic 2 (see also [11]). In this paper we give necessary and sufficient conditions for AG to be tame where A is an arbitrary algebra and G a finite group. In order to state the main theorem we need some notationa. For any positive integer n we will denote by Xn the quiver and by Yn the oriented cycle J.n
,J.n-1/n y
n
n-1
.•
~1 2
l
i
•
For
')1
•
••
•
5 and any sequence of positive integers n1 , ••• ,n 8 satisfying the conditions: s ) 2, n1 > 1, n8 < n, and n~
ni + 1
< "i+1 '
i
= 1, ••• ,s-1,
we will denote by A"(
the bounden quiver algebra I
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