Modules And Comodules Of Quantum Algebras

The quantum algebra is an algebra with generators and relations. Let R = Z[v]_m, where v is an indeterminate and m is an ideal in Z[v] generated by v-1 and a fixed oddprime p. R' is the fraction of R. U' is a quantum algebra over R' associated to Cartan matrix (a_ij) .U is a subquantum algebra over R of U' .If definedcomultiplication, counit and antipode, U' is an R' -Hopf algebra. U is also R-Hopf algebra with related structure. In reference [1] the authors introduce the quantum coordinate algebra R[U] = F_δ((?)(R)) as a suitably dual coalgebra of U. In this paper some important propositions about quantum algebra will be in detail proved in such ways which are essential. Algebras and coalgebras are the dual. As usual, if let k be a real, and C be a k coalgebra, C^* =hom_k(C,k) is a k-algebra with relatedstructure .Dually, let A be a k-algebra, and we can get a subspace of A^*, i.e. A⁰ =R_k(A_m)∩A^* which is k-algebra, and if A is k-Hopf algebra, so is A⁰. In this paper we will think about their relations one another. Algebra Modules and coalgebra comodules are also dual. We research modules as important as do comodules. In this paper we research their duality, and give some generalizations. The rationality is vital to researching infinite dimensional algebras, so in this paper we will make some discussion of quantum algebra modules and coalgebra comodules.