Quantum Gordan Formula And Combinatorial Problems

For any Hopf algebra H , it is one of the most important goals that we discuss decomposition and commutation for the tensor product of two H - modules in Hopf algebra theory . In [1] , C.Kassel proved that if H is a braided Hopf algebra(or quasi-triangular Hopf algebra), then for any two H -modules V,W , there exists a module isomorphismwhere R H H is a universal R -matrix , and the braided condition is used. However, as we know, for any Hopf algebra H , the tensor product of two H -modules is not always commutative.In Chapter One,we introduce the background of Hopf algebra and our research goals, present the basic properties of Hopf algebra which we need in our paper.Chapter Two is a basis of the whole article. First, we prove the basic theorem(Theorem 2.1) using mathematical induction on n, m .Theorem 2.1 Let,4 be a associative algebra , and {e0,e1, ...,en,...} be a set of linearly independent elements in A , which satisfiesfor all n N , where e-1 = 0 . Then (l)forall n m,(2) As an algebra, k[e1 ] = k[x], where k[x] is a polynomial algebra on field k,k[e1] is a subalgebra of A generated bye1. Second,we use computer programs Mathematica 4.0 to find rules, then we useinduction to figure out a standard basis satisfying formula (2.1) for polynomial algebra k[x],Chapter Three is the important part of this article. Beginning with Grothendieck ring,we prove theorem 3.1:Theorem 3.1 Let H be any Hopf algebra, and its finite dimensional modules be commutative,K(H) be its Grothendieck ring. If e0e0 = e0,ene1 = e1en - en+1 + en-1 inK(H), then for all where en =[Mn] is an isomorphic class of finite-dimensional simple module Mn.So we can present the conditions when the tensor product of two finite-dimensional semisimple modules for any Hopf algebra commutes. Then we fix H = U (sl(2)) orU(sl(2)) ,we can get the unified proof of the Clebsch-Gordan formula and quantumClebsch-Gordan formula, and tensor product of any two finite-dimensional semisimple H -modules is commutative. But in our proof the braided condition is not used .In Chapter Four,we start from special case, and work out A'a,B'n (0 n 10). Then using induction on n, i ,we get two combination formula A'n, B'n and their relations, with which we can decompose V(1) 2n,V(1) (2n+1) into direct sum of simple modules.In Chapter Five ,we give the explicit decomposition of V(n)m.In this article, using computer program,we look for the combination laws, and hence we get nice results.