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

The gauge action of the Lie group $G$ on the chiral WZNW phase space ${\cal M}_{\check G}$ofquasiperiodicfieldswith$\check G$-valuedmonodromy,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 ${\mathcal{M}}_{\check{G}}$ 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 $\mathrm{\Omega }\in G^{\ast }$. 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.