Twisting the $q$-deformations of compact semisimple Lie groups

Abstract

Given a compact semisimple Lie group $G$ of rank $r$, and a parameter $q$ > 0, we can define new associativity morphisms in ${\rm Rep}(G_q)$ using a $3$-cocycle $\Phi$ on the dual of the center of $G$, thus getting a new tensor category ${\rm Rep}(G_q)^\Phi$. For a class of cocycles $\Phi$ we construct compact quantum groups $G^\tau_q$ with representation categories ${\rm Rep}(G_q)^\Phi$. 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 $SU_q(2)$. More generally, for $G=SU(n)$, we get quantum group realizations of the Kazhdan–Wenzl categories.