Solution To The Clebsch-Gordan Problem For Selfinjective Algebras Of Finite Representation Type

The Clebsch-Gordan problem is the following:Given two indecomposable representations of a group G,how to decompose their tensor products into a direct sum of indecomposables.However,the Clebsch-Gordan problem can be posed for any Krull-Schmidt category equipped with a tensor productIn this paper,we mainly discuss the representation categories of selfinjective algebras of finite representation type.these algebras which are quotient of path algebras.We first prove that the representation categories of some of these self-injective algebras can be equipped with tensor products defined point-wise and arrow-wise.Then we solve the Clebsch-Gordan problem for those selfinjective algebras explicitlyIn chapter 2.we study Nakayama selfinjective algebras.We first prove that the representation category of these algebras can be equipped with tensor products defined point-wise and arrow-wise.Then we explicitly calculate the tensor product of two indecomposable representations of a Nakayama selfinjective algebra and decompose the tensor products into a direct sum of indecomposables.Thus we solve the Clebsch-Gordan problem for Nakayama selfinjective algebrasIn chapter 3.we prove that the representation category of the trivial exten?sion T(KQA)of path algebras Dynkin type A can also be equipped with tensor products defined point-wise and arrow-wise.Then we solve the Clebsch-Gordan problem for T(KQA)Based on the results of chapter 3.in chapter 4 we solve the Clebsch-Gordan problem for the trivial extension of path algebras of Dynkin type D.