The non-Abelian momentum map for Poisson-Lie symmetries on the chiral WZNW phase space

TL;DR

This paper develops a non-Abelian momentum map for Poisson-Lie symmetries in the chiral WZNW phase space, establishing a correspondence between monodromy variables and dual group elements, and generalizing the classical dynamical Yang-Baxter equation.

Contribution

It introduces an explicit momentum map for Poisson-Lie symmetries on the WZNW phase space and relates monodromy variables to dual group elements, extending the classical dynamical Yang-Baxter equation.

Findings

Explicit correspondence between monodromy and dual variables.

Conversion of PL groupoid to a canonical form using the Heisenberg double.

Natural PL generalization of the classical dynamical Yang-Baxter equation.

Abstract

The gauge action of the Lie group $G$ on the chiral WZNW phase space ${\cal M}_{\check G}$ of quasiperiodic fields with $\check G$-valued monodromy, where $\check G\subset G$ is an open submanifold, is known to be a Poisson-Lie (PL) action with respect to any coboundary PL structure on $G$, if the Poisson bracket on ${\cal M}_{\check G}$ is defined by a suitable monodromy dependent exchange $r$-matrix. We describe the momentum map for these symmetries when $G$ is either a factorisable PL group or a compact simple Lie group with its standard PL structure. The main result is an explicit one-to-one correspondence between the monodromy variable $M \in \check G$ and a conventional variable $\Omega \in G^*$. This permits us to convert the PL groupoid associated with a WZNW exchange $r$-matrix into a `canonical' PL groupoid constructed from the Heisenberg double of $G$, and consequently to obtain a natural PL generalization of the classical dynamical Yang-Baxter equation.