Twisting of the Quantum double and the Weyl algebra

TL;DR

This paper explores how the quantum double construction can be twisted using an R-matrix, revealing a connection to the Weyl algebra and proposing methods to extend these ideas to locally compact quantum groups.

Contribution

It introduces a twisting of the quantum double and its dual, linking the twisted dual to the Weyl algebra, and discusses potential extensions to locally compact quantum groups.

Findings

Twisting the quantum double relates to the Weyl algebra.

The twisted dual of D(G) is isomorphic to the Weyl algebra.

Proposes approaches for extending to locally compact quantum groups.

Abstract

Quantum double construction, originally due to Drinfeld and has been since generalized even to the operator algebra framework, is naturally associated with a certain (quasitriangular) $R$-matrix ${\mathcal R}$. It turns out that ${\mathcal R}$ determines a twisting of the comultiplication on the quantum double. It then suggests a twisting of the algebra structure on the dual of the quantum double. For $D(G)$, the $C^*$-algebraic quantum double of an ordinary group $G$, the "twisted $\hat{D(G)}$" turns out to be the Weyl algebra $C_0(G)\times_{\tau}G$, which is in turn isomorphic to ${\mathcal K}(L^2(G))$. This is the $C^*$-algebraic counterpart to an earlier (finite-dimensional) result by Lu. It is not so easy technically to extend this program to the general locally compact quantum group case, but we propose here some possible approaches, using the notion of the (generalized) Fourier transform.