Dual Affine Quantum Groups

Abstract

Let $\hat{\mathfrak{g}}$ be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let $\hat{\mathfrak{h}}$ be the dual Lie bialgebra. By dualizing the quantum double construction - via formal Hopf algebras - we construct a new quantum group $U_q(\hat{\mathfrak{h}})$, dual of $U_q(\hat{\mathfrak{g}})$. Studying its restricted and unrestricted integer forms and their specializations at roots of 1 (in particular, their classical limits), we prove that $U_q(\hat{\mathfrak{h}})$ yields quantizations of $\hat{\mathfrak{h}}$ and ${\hat{G}}^{\infty }$ (the formal group attached to $\hat{\mathfrak{g}}$), and we construct new quantum Frobenius morphisms. The whole picture extends to the untwisted affine case the results known for quantum groups of finite type.

36 pages, AMS-TeX file. This the author's final version, corresponding to the pronted journal version. arXiv admin note: text overlap with arXiv:q-alg/9511022

References in corpus (2)

• Quantization of Lie bialgebras, V
• The quantum duality principle

Cited by in corpus (3)

• The global quantum duality principle
• Braiding structures on formal Poisson groups and classical solutions of the QYBE
• The global quantum duality principle: a survey through examples