On Automorphisms and Universal R-Matrices at Roots of Unity

TL;DR

This paper investigates conditions under which different quotients of quantum groups at roots of unity are equivalent, focusing on the existence and transformation of universal R-matrices, especially for U_q(sl(2)).

Contribution

It establishes that quotient equivalence and R-matrix transformation occur only when q^4=1, providing explicit automorphisms and R-matrix expressions.

Findings

Equivalence of quotients occurs only when q^4=1.

Universal R-matrices can be transformed between quotients under specific conditions.

Explicit formulas for automorphisms and R-matrices at q^4=1.

Abstract

Invertible universal R-matrices of quantum Lie algebras do not exist at roots of unity. There exist however quotients for which intertwiners of tensor products of representations always exist, i.e. R-matrices exist in the representations. One of these quotients, which is finite dimensional, has a universal R-matrix. In this paper, we answer the following question: on which condition are the different quotients of U_q(sl(2)) (Hopf)-equivalent? In the case when they are equivalent, the universal R-matrix of one can be transformed into a universal R-matrix of the other. We prove that this happens only when q^4=1, and we explicitly give the expressions for the automorphisms and for the transformed universal R-matrices in this case.