Representations And Filtrations For Quantum Algebras

Let A denote the local ring z[v]m where v is an indeterminate and m is the ideal generated by v-1 and an odd prime p. The fraction of A is denoted by A' = Q(v). We can define a quantum algebra as U' which is generated byEi, Fi,Ki+1, i = 1,2,n under some relations. U is the ,4-subalgebra of U generated by Ei(N), Fi(N),Ki, Ki-1, i =1,2, n.Let X=Znbe the set of weights and X+ the set of dominant w weights. Let w0 be the longest element of W, where W is the Weyl group of R. In this paper, we prove some isomorphism theorems of dual module of integrable U -module E, and functor D is a right covariant functor. In this paper, we prove that if U-module M has a D filtration then M has a good filtration and if U-module N has a good filtration then Nt has a D filtration. Furthermore, we also prove if then good filtration. In fact we prove has a D filtration.