Quantum double of Uq((sl2)⩽0)

Let Uq(sl2) be the quantized enveloping algebra associated to the simple Lie algebra sl2. In this paper, we study the quantum double Dq of the Borel subalgebra Uq((sl2)⩽0) of Uq(sl2). We construct an analogue of Kostant–Lusztig Z[v,v−1]-form for Dq and show that it is a Hopf subalgebra. We prove that, over an algebraically closed field, every simple Dq-module is the pull-back of a simple Uq(sl2)-module through certain surjection from Dq onto Uq(sl2), and the category of finite-dimensional weight Dq-modules is equivalent to a direct sum of |k×| copies of the category of finite-dimensional weight Uq(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.