The $R$--matrix action of untwisted affine quantum groups at roots of 1

Abstract

Let $\hat{\mathfrak{g}}$ be an untwisted affine Kac-Moody algebra. The quantum group $U_h(\hat{\mathfrak{g}})$ (over $\mathbb{C}[[h]]$) is known to be a quasitriangular Hopf algebra: in particular, it has a universal $R$--matrix, which yields an $R$--matrix for each pair of representations of $U_h(\hat{\mathfrak{g}})$. On the other hand, the quantum group $U_q(\hat{\mathfrak{g}})$ (over $\mathbb{C}(q)$) also has an $R$--matrix for each pair of representations, but it has not a universal $R$--matrix so that one cannot say that it is quasitriangular. Following Reshetikin, one introduces the (weaker) notion of braided Hopf algebra: then $U_q(\hat{\mathfrak{g}})$ is a braided Hopf algebra. In this work we prove that also the unrestricted specializations of $U_q(\hat{\mathfrak{g}})$ at roots of 1 are braided: in particular, specializing $q$ at 1 we have that the function algebra $F\left[\hat{H}\right]$ of the Poisson proalgebraic group $\hat{H}$ dual of $\hat{G}$ (a Kac-Moody group with Lie algebra $\hat{\mathfrak{g}}$) is braided. This is useful because, despite these specialized quantum groups are not quasitriangular, the braiding is enough for applications, mainly for producing knot invariants. As an example, the action of the $R$--matrix on (tensor products of) Verma modules can be specialized at odd roots of 1.

12 pages, AMS-TeX C, Version 2.1c - this is the author's file of the final version (after the refereeing process), as sent for publication

References in corpus (2)

• A PBW basis for Lusztig's form of untwisted affine quantum groups
• Braidings of Poisson groups with quasitriangular dual (Tressages des groupes de Poisson à dual quasitriangulaire)

Cited by in corpus (2)

• Tressages des groupe de Poisson formels à dual quasitriangulaire
• Braiding structures on formal Poisson groups and classical solutions of the QYBE