Quantum double of ${\mathrm{U}}_q({({\mathfrak{sl}}_2)}^{⩽0})$

Abstract

Let ${\mathrm{U}}_q({\mathfrak{sl}}_2)$ be the quantized enveloping algebra associated to the simple Lie algebra ${\mathfrak{sl}}_2$. In this paper, we study the quantum double ${\mathrm{D}}_q$ of the Borel subalgebra ${\mathrm{U}}_q({({\mathfrak{sl}}_2)}^{⩽0})$ of ${\mathrm{U}}_q({\mathfrak{sl}}_2)$. We construct an analogue of Kostant–Lusztig $\mathbb{Z}[v,v^{\mathrm{-1}}]$-form for ${\mathrm{D}}_q$ and show that it is a Hopf subalgebra. We prove that, over an algebraically closed field, every simple ${\mathrm{D}}_q$-module is the pull-back of a simple ${\mathrm{U}}_q({\mathfrak{sl}}_2)$-module through certain surjection from ${\mathrm{D}}_q$ onto ${\mathrm{U}}_q({\mathfrak{sl}}_2)$, and the category of finite-dimensional weight ${\mathrm{D}}_q$-modules is equivalent to a direct sum of $|k^{\times }|$ copies of the category of finite-dimensional weight ${\mathrm{U}}_q({\mathfrak{sl}}_2)$-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.