On the centers of quantum groups of An-type

Abstract

Let g be the finite dimensional simple Lie algebra of type An, and let U = U q (g,Λ) and U = U q (g,Q) be the quantum groups defined over the weight lattice and over the root lattice, respectively. In this paper, we find two algebraically independent central elements in U for all n ≥ 2 and give an explicit formula of the Casimir elements for the quantum group U = U q (g,Λ), which corresponds to the Casimir element of the enveloping algebra U(g). Moreover, for n = 2 we give explicitly generators of the center subalgebras of the quantum groups U = U q (g,Λ) and U = U q (g,Q).