Properties Of Indecomposable Modules On Generalized Taft Algebra And Lie Algebra Sl(2,K)

In 1908, J.H.M.Wedderburn proposed semisimple algebra representation theory. The theory provides a new way for the study of algebraic structures. In 1929, E.Nother based on this theory, put forward the semisimple algebra representation of the module theory,expanding the application of the theory. The representation theory mainly studies how to"represent" the elements in the algebraic structure into linear transformations in vector spaces or matrices in full matrix rings. It would be easy for the researchers to analyze the properties of algebraic structures.Generalized Taft algebra is one of important Hopf algebra, with both non-commutative and non-cocommutative characteristics. It is of great significance to study the algebra representation of the generalized Taft algebra. Based on the principle of representation theory,this paper mainly studies the indecomposable modules of generalized Taft algebra under the adjoint action, and analyzes its basic properties.This paper is organized as follows:In the first chapter, we review the related concepts and definitions, including Lie algebra,weight space, maximal vector, bialgebra, Hopf algebra, adjoint action, etc., which are the preparation for the subsequent chapters. The basic conclusions of the paper are given, such as the representation of the Taft algebra indecomposable module, the representation of the generalized Taft algebra up to isomorphism.In the second chapter, we discuss the representation of three-dimensional simple Lie algebra A1? sl(2,K)=L,and deduce the process of solving the rank of the indecomposable module. Firstly, according to the irreducible representation of the simple Lie algebra, we get the matrix corresponding to the irreducible representation of arbitrary element ax+by+cz.Secondly, according to the matrix corresponding to the base element, determine the image of the element in the irreducible representation, and then compute its rank. Finally, we can get a vector space V through the calculation if and only if dimV=2, c2+ab=0,the vector spaceV is indecomposable module and the rank(V) = 1.In the third chapter, we mainly discuss the indecomposable module of generalized Taft algebra under the concomitant action. Firstly, the relationship between generators of the Taft algebra is obtained. Secondly, we can obtain indecomposable modules by derivation,including [gi] and [gn-1hj] generated by base elements. Finally, we can transform the generalized Taft algebra into the direct sum (?)...