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

Abstract

The gauge action of the Lie group G on the chiral Wess-Zumino-Novikov-Witten (WZNW) phase space ℳ Ǧ of quasiperiodic fields with Ǧ-valued monodromy, where Ǧ ⊂ 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 ℳ Ǧ is defined by a suitable monodromy-dependent exchange r-matrix. We describe the momentum map for these symmetries when G is either a factorizable 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 ∈ Ǧ and a conventional variable Ω ∈ 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.