Quantization of Poisson Groups

Abstract

Let $G^{\tau }$ be a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfel'd structure of Poisson group, let $H^{\tau }$ be its dual Poisson group. By means of quantum double construction and dualization via formal Hopf algebras, we construct new quantum groups $U_{q,\phi }^M(\mathfrak{h})$ --- dual of $U_{q,\phi }^{M^{\prime }}(\mathfrak{g})$ --- which yield infinitesimal quantization of $H^{\tau }$ and $G^{\tau }$, we study their specializations at roots of 1 (in particular, their classical limits), thus discovering new quantum Frobenius morphisms. The whole description dualize for $H^{\tau }$ what was known for $G^{\tau }$, completing the quantization of the pair $(G^{\tau },H^{\tau })$.

AMS-TeX file, 45 pages, final author's version of the printed paper

References in corpus (1)

• Quantum function algebras as quantum enveloping algebras

Cited by in corpus (5)

• Differential operators on quantized flag manifolds at roots of unity II
• Braiding structures on formal Poisson groups and classical solutions of the QYBE
• Twisted deformations vs. cocycle deformations for quantum groups
• F_q[M_n], F_q[GL_n] and F_q[SL_n] as quantized hyperalgebras
• Formal multiparameter quantum groups, deformations and specializations