On the Drinfel'd-Kohno Equivalence of Groups and Quantum Groups

TL;DR

This paper presents a method for calculating matrix representations of the twist element in quantum groups, explores the crystal limit, and offers a new interpretation of q-deformation via tensor products of Lie group representations.

Contribution

It introduces a practical method for computing F-matrices, analyzes their behavior in the crystal limit, and provides a novel perspective on q-deformation in relation to Lie group representations.

Findings

F-matrices have a crystal limit as q approaches 0

F-matrices from 0 to q are simpler than from 1 to q

Q-deformation can be interpreted through tensor products of finite-dimensional Lie group representations

Abstract

A method to calculate matrix representations of the twist element $\ff$ of Drinfel'd -- chosen to be unitary -- is given and illustrated at some examples. It is observed that for these F-matrices the crystal limit $q\!\to\! 0$ exists and that \mbox{F-matrices} twisting from $0$ to $q$ are of a simpler form than F-matrices twisting from $1$ to $q$. These results lead to a new interpretation of $q$-deformation in terms of tensor products of finite-dimensional representations of compact simple Lie groups.