Modular Double of Quantum Group

TL;DR

The work introduces a modular double for quantum groups by pairing $q$ with its modular dual $\tilde{q}$ and adjoining dual Weyl-type generators to define a logarithm-like element. It constructs a universal $R$-matrix valid for both $q$ and $\tilde{q}$ as a product of an exponential factor and a dualizing function $\psi(p)$, avoiding small-denominator issues. An explicit construction for $SL_q(2)$ sits inside a doubled algebra $\mathcal{D}=\mathcal{C}_q\otimes\mathcal{C}_{\tilde{q}}$, with canonical generators realized via $p_n$ and a pentagon-consistent $\psi(p)$. This framework points to hidden symmetries in conformal field theory and related mathematical structures, and it opens questions for generalization to other groups and rigorous closure definitions.

Abstract

An extension of Quantum Group is described. We propose to unite the quantum groups with parameter q and with parameter modularly dual to q.