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.

Abstract. We describe all finite-dimensional algebras A over an algebraically closed field for which the algebra T 2 (A) of 2 × 2 upper triangular matrices over A is of tame representation type. Moreover, the algebras A for which T 2 (A) is of polynomial growth (respectively, domestic, of finite representation type) are also characterized.

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Abstract. We prove that a completely separating incidence algebra of a partially ordered set is of tame representation type if and only if the associated Tits integral quadratic form is weakly non-negative.

Abstract. With the help of Galois coverings, we describe the tame tensor products A ⊗ K B of basic, connected, nonsimple, finite-dimensional algebras A and B over an algebraically closed field K. In particular, the description of all tame group algebras AG of finite groups G over finite-dimensional algebras A is completed.Introduction. Throughout the paper K will denote a fixed algebraically closed field. By an algebra we mean a finite-dimensional K-algebra (associative, 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 = Q A 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 finite number of discrete and a finite number of one-parameter families. The second class is formed by the wild algebras whose representation theory comprises the representation theories of all finite-dimensional algebras over K. Accordingly, we may realistically hope to classify the indecomposable finitedimensional 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 product A ⊗ K B of two nonsimple algebras A and B is tame. The class of tensor product algebras contains several important classes of algebras, including: