Twisting the q-deformations of compact semisimple Lie groups

TL;DR

This paper introduces new quantum groups derived from compact semisimple Lie groups by twisting their representation categories with 3-cocycles, leading to novel algebraic structures and connections to known quantum groups.

Contribution

It constructs new compact quantum groups using 3-cocycle twists on the representation categories of classical Lie groups, generalizing known quantum groups like SU_{-q}(2).

Findings

Constructed quantum groups G_q^τ for various Lie groups G.

Reproduced Woronowicz's SU_{-q}(2) as a special case.

Realized Kazhdan-Wenzl categories through this twisting approach.

Abstract

Given a compact semisimple Lie group $G$ of rank $r$, and a parameter $q>0$, we can define new associativity morphisms in Rep(Gq) using a 3-cocycle $\Phi$ on the dual of the center of G, thus getting a new tensor category Rep(Gq)$^\Phi$. For a class of cocycles $\Phi$ we construct compact quantum groups $G^\tau_q$ with representation categories Rep(Gq)$^\Phi$. The construction depends on the choice of an r-tuple $\tau$ of elements in the center of G. In the simplest case of G=SU(2) and $\tau=-1$, our construction produces Woronowicz's quantum group SU_{-q}(2) out of SUq(2). More generally, for G=SU(n), we get quantum group realizations of the Kazhdan-Wenzl categories.