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

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

1998-05-04

arxiv.org/abs/math/9805009v3

J. Pure Appl. Algebra 155 (2001), no. 1, 41-52

Mathematics

Quantum Algebra

12 pages, AMS-TeX C, Version 2.1c. To appear in: "Journal of Pure and Appllied Algebra"

Scientific paper

Let $\hat{\frak g}$ be an untwisted affine Kac-Moody algebra. The quantum group $U_h(\hat{\frak g})$ (over $C[[h]]$) 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 $U_h(\hat{\frak g})$. On the other hand, the quantum group $U_q(\hat{\frak g})$ (over $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{\frak g})$ is a braided Hopf algebra. In this work we prove that also the unrestricted specializations of $U_q(\hat{\frak g})$ at roots of 1 are braided: in particular, specializing q at 1 we have that the function algebra $F \big[ \hat{H} \big]$ of the Poisson proalgebraic group $\hat{H}$ dual of $\hat{G}$ (a Kac-Moody group with Lie algebra $\ghat$) 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.

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