Tame tensor products of algebras

Abstract

With the help of Galois coverings, we describe the tame tensor products A K B of basic, connected, nonsimple, nite-dimensional algebras A and B over an algebraically closed eld K. In particular, the description of all tame group algebras AG of nite groups G over nite-dimensional algebras A is completed. Introduction. Throughout the paperK will denote a xed algebraically closed eld. By an algebra we mean a nite-dimensional K-algebra (associa- tive, with an identity) which we moreover assume to be basic and connected. An algebra A can be written as a bound quiver algebra A= KQ=I, where Q = QA is the Gabriel quiver of A and I is an admissible ideal in the path algebra KQ of Q. By Drozd's Tame and Wild Theorem (9) the class of algebras may be divided into two disjoint classes. One class consists of the tame algebras for which the indecomposable modules occur, in each dimension d, in a - nite number of discrete and a nite number of one-parameter families. The second class is formed by the wild algebras whose representation theory com- prises the representation theories of all nite-dimensional algebras over K. Accordingly, we may realistically hope to classify the indecomposable nite- dimensional modules only for the tame algebras. The representation theory of arbitrary tame algebras is still only emerging. We are concerned with the problem of describing when the tensor prod- uct A KB of two nonsimple algebras A and B is tame. The class of tensor product algebras contains several important classes of algebras, including: (1) the group algebrasAG= A KKG of nite groupsG with coecien ts in algebras A;