On moment maps associated to a twisted Heisenberg double

TL;DR

This paper explores the structure of moment maps in twisted Heisenberg doubles, clarifying Poisson-Lie symmetries and reductions, and introduces a non-anomalous moment map enabling gauging of deformed WZW models.

Contribution

It provides a clear framework for Poisson-Lie symmetries in twisted Heisenberg doubles and constructs a non-anomalous moment map for quasi-adjoint actions.

Findings

Demonstrates that group actions realize Poisson-Lie symmetries.

Introduces the concept of Poisson-Lie subsymmetry and reduction.

Constructs a non-anomalous moment map for gauging models.

Abstract

We review the concept of the (anomalous) Poisson-Lie symmetry in a way that emphasises the notion of Poisson-Lie Hamiltonian. The language that we develop turns out to be very useful for several applications: we prove that the left and the right actions of a group $G$ on its twisted Heisenberg double $(D,\kappa)$ realize the (anomalous) Poisson-Lie symmetries and we explain in a very transparent way the concept of the Poisson-Lie subsymmetry and that of Poisson-Lie symplectic reduction. Under some additional conditions, we construct also a non-anomalous moment map corresponding to a sort of quasi-adjoint action of $G$ on $(D,\kappa)$. The absence of the anomaly of this "quasi-adjoint" moment map permits to perform the gauging of deformed WZW models.