Quantum double of Uq ((sl2)≤0)

Let U-q (sl(2)) be the quantized enveloping algebra associated to the simple Lie algebra sl(2). In this paper, we study the quantum double D-q of the Borel subalgebra U-q ((sl(2))(<= 0)) of U-q (sl(2)). We construct an analogue of Kostant-Lusztig Z[v, v(-1)]-form for D-q and show that it is a Hopf subalgebra. We prove that, over an algebraically closed field, every simple D-q-module is the pull-back of a simple Uq(sl(2))-module through certain surjection from Dq onto U-q(sl(2)), and the category of finite-dimensional weight D-q-modules is equivalent to a direct sum of vertical bar k(x)vertical bar copies of the category of finite-dimensional weight Uq (sl(2))-modules. As an application, we recover (in a conceptual way) Chen's results [H.X. Chen, Irreducible representations of a class of quantum doubles, J. Algebra 225 (2000) 391-409] as well as Radford's results [D.E. Radford, On oriented quantum algebras derived from representations of the quantum double of a finite-dimensional Hopf algebras, J. Algebra 270 (2003) 670-695] on the quantum double of Taft algebra. Our main results allow a direct generalization to the quantum double of the Borel subalgebra of the quantized enveloping algebra associated to arbitrary Cartan matrix. (C) 2007 Elsevier Inc. All rights reserved.