FINITE-DIMENSIONAL REPRESENTATION OF QUANTUM GROUP Uq(f(K))

Abstract

Let q be not a root of unity, for any Laurent polynomial f(K) in k[K,K−1]. In this paper, we define an algebra Uq(f(K)) and prove that Uq(f(K)) is Noetherian and has no zero divisors. We also give the necessary and sufficient condition for Uq(f(K)) to be a Hopf algebra. Finally, all finite-dimensional simple Uq(f(K))-modules and the centre Zq(f(K)) of Uq(f(K)) are given.