Direct And Decomposition Of A Kind Of Tensor

Hopf algebra is an important research field of algebras. The construction and classification of Hopf algebras play an important role in the theory of Hopf algebras. During the last few years several classification results for pointed Hopf algebras were obtained based on the theory of Nichols algebras. On the other hand, as the new Hopf algebras appear, the representation theory of pointed Hopf algebras attracts many mathematicians’attention, and many interesting results have been gotten.Krop and Radford defined the rank as a measure of complexity for Hopf algebras. They classified all finite dimensional pointed Hopf algebras of rank one over an algebraically field of characteristic 0. Scherotzke classified such Hopf algebras over an algebraically field of positive characteristic. Wang, You and Chen generalized the result to the case of an arbitrary field without any restriction. They showed that a pointed Hopf algebra of rank one over an arbitrary field is isomorphic to a quotient of a Hopf-Ore extension of its coradical. Furthermore, they studied the representation theories of Hopf-Ore extensions of group algebras and pointed Hopf algebras of rank one, and classified finite dimensional indecomposable weight modules over such Hopf algebras. In this paper, based on the above result, we study the tensor product of two weight modules, and decompose such tensor products into the direct sum of indecomposable modules, here we only consider the case that the base field is an algebraically closed field of characteristic zero.We organize the paper as follows. In Section 1, we present some basic definitions, notations and the structures of the Hopf-Ore extension of group algebras. In Section 2, under the hypothesis that the base field is an algebraically closed field, we introduce the finite dimensional indecomposable weight modules over the Hopf-Ore extension H= kG(x-1,a,0) of a group algebra, and give the structures and classification of these modules. In Section 3, under the hypothesis that the base field is an algebraically closed field of characteristic zero, we study the tensor products of the modules given in Section 2. Note that the order of x is either infinite or finite. In the case that the order of x is infinite, we give the decompositions of the tensor products of any two finite dimensional indecomposable weight modules into the direct sum of indecomposable modules. In the case that the order of x is finite, the finite dimensional indecomposable weight modules can be clarified into two types:nilpotent type and non-nilpotent type. We first investigate the tensor product of a module of nilpotent type with one of non-nilpotent type, and decompose such a tensor product into the direct sum of indecomposable modules. It is shown that any indecomposable summand of such a tensor product is of non-nilpotent type. Then we investigate the tensor product of two modules of non-nilpotent type. The decomposition of such a tensor product into the direct sum of indecomposable modules is given, which shows that the summands of such a tensor product are either all of nilpotent type, or all of non-nilpotent type. Finally, we investigate the tensor product of two modules of nilpotent type. It is shown that any indecomposable summand of such a tensor product is of nilpotent type, and that the number of summands in any decomposition of such a tensor product is equal to the lesser of the dimensions of the two module in the tensor product. Furthermore, when one of the two modules has dimension not exceeding the order of x, we decompose such a tensor product into the direct sum of indecomposable modules.