A Heisenberg double addition to the logarithmic Kazhdan--Lusztig duality

TL;DR

This paper explores the structure of the Heisenberg double as a module algebra over the Drinfeld double in the context of logarithmic conformal field theories and quantum groups, providing explicit examples with the Taft Hopf algebra.

Contribution

It introduces a new perspective on the Heisenberg double as a module algebra over the Drinfeld double, linking it to quantum-group field algebras in logarithmic CFTs, with explicit constructions for U_q(sl(2)).

Findings

Heisenberg double H(B*) can be structured as a D(B)-module algebra.

Explicit example with Taft Hopf algebra related to U_q(sl(2)).

H_qsl(2) has a specific algebraic form involving q-deformed relations.

Abstract

For a Hopf algebra B, we endow the Heisenberg double H(B^*) with the structure of a module algebra over the Drinfeld double D(B). Based on this property, we propose that H(B^*) is to be the counterpart of the algebra of fields on the quantum-group side of the Kazhdan--Lusztig duality between logarithmic conformal field theories and quantum groups. As an example, we work out the case where B is the Taft Hopf algebra related to the U_qsl(2) quantum group that is Kazhdan--Lusztig-dual to (p,1) logarithmic conformal models. The corresponding pair (D(B),H(B^*)) is "truncated" to (U_qsl(2),H_qsl(2)), where H_qsl(2) is a U_qsl(2) module algebra that turns out to have the form H_qsl(2)=\oC_q[z,d]\tensor C[\lambda]/(\lambda^{2p}-1), where C_q[z,d] is the U_qsl(2)-module algebra with the relations z^p=0, d^p=0, and d z = q-q^{-1} + q^{-2} zd.