On moment maps associated to a twisted Heisenberg double

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.

52 pages, LaTeX, introduction substantially enlarged, several explanatory remarks added, final published version

References in corpus (3)

• Boundary WZW, G/H, G/G and CS theories
• Poisson-Lie symmetry and q-WZW model
• Quasitriangular chiral WZW model in a nutshell

Cited by in corpus (7)

• Poisson-Lie interpretation of trigonometric Ruijsenaars duality
• Poisson-Lie analogues of spin Sutherland models
• Poisson-Lie generalization of the Kazhdan-Kostant-Sternberg reduction
• Reduction of a bi-Hamiltonian hierarchy on $T^{\ast }\mathrm{U}(n)$ to spin Ruijsenaars--Sutherland models
• Poisson reductions of master integrable systems on doubles of compact Lie groups
• Poisson-Lie analogues of spin Sutherland models revisited
• u-Deformed WZW Model and Its Gauging