On the representation type of tensor product algebras

Abstract

The representation type of tensor product algebras of finite-dimensional algebras is considered. The characterization of algebras A, B such that A⊗B is of tame representation type is given in terms of the Gabriel quivers of the algebras A, B. Introduction. In this paper by an algebra we mean a finite-dimensional algebra over a fixed algebraically closed field K. All algebras are assumed to be basic indecomposable with respect to the direct product. Our aim is to determine the representation type of the tensor product algebra B⊗K C of two algebras B and C in terms of the quivers with relations describing the algebras B and C. One of the motivations for our study is to introduce a unified approach to the investigation of the representation type of several important classes of algebras including: (i) The group algebras B[G] of a finite group G with coefficients in an algebra B (studied in [MS, S1]). (ii) The lower triangular n× n matrix algebras (0.1) Tn(B) =   B 0 . . . 0 B B . . . 0 .. .. . . . B B . . . B   with n ≥ 2 and with coefficients in an algebra B (studied in [AR, Br2, L1, L2, LS, S2]). (iii) The factor algebras Tn,r(B) := Tn(B)/J r n(B) of Tn(B), n, r ≥ 2, studied in [HM], where Jn(B) the ideal of strictly lower triangular n × n matrices. (iv) The path algebra BQ of a bound quiver Q with coefficients in an algebra B. 1991 Mathematics Subject Classification: Primary 16G60.