Quantum double of U_q ((sl₂)^{≤ 0})

Let U_q (sl₂) be the quantized enveloping algebra associated to the simple Lie algebra sl₂. In this paper, we study the quantum double D_q of the Borel subalgebra U_q ((sl₂)^{≤ 0}) of U_q (sl₂). 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 U_q (sl₂)-module through certain surjection from D_q onto U_q (sl₂), and the category of finite-dimensional weight D_q-modules is equivalent to a direct sum of | k^× | copies of the category of finite-dimensional weight U_q (sl₂)-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.