Quantum double of ${\rm U}_q((\ksl_2)^{\leq 0})$
| dc.creator | Hu, Jun | |
| dc.creator | Zhang, Yinhuo | |
| dc.date | 2005-12-25 | |
| dc.date | 2007-09-19 | |
| dc.date.accessioned | 2026-07-07T08:30:38Z | |
| dc.date.available | 2026-07-07T08:30:38Z | |
| dc.description | 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 ${U}_q((sl_2)^{\leq 0})$ 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 pullback of a simple ${U}_q(sl_2)$-module through certain surjection from $D_q$ onto ${U}_q(sl_2)$, and the category of finite dimensional weight $D_q$-modules is equivalent to a direct sum of $|k^{\times}|$ copies of the category of finite dimensional weight ${U}_q(sl_2)$-modules. As an application, we recover (in a conceptual way) Chen's results as well as Radford's results 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. | |
| dc.identifier | https://arxiv.org/abs/math/0512563 | |
| dc.identifier | http://arxiv.org/abs/math/0512563 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/138285 | |
| dc.subject | Quantum Algebra | |
| dc.subject | Representation Theory | |
| dc.title | Quantum double of ${\rm U}_q((\ksl_2)^{\leq 0})$ | |
| dc.type | text |