The Chiral WZNW Phase Space and its Poisson-Lie Groupoid

TL;DR

This paper establishes a detailed relationship between monodromy-dependent 2-forms and exchange r-matrices in the chiral WZNW model, revealing new dynamical Yang-Baxter equations and their interpretation via Poisson-Lie groupoids.

Contribution

It introduces a new dynamical generalization of the classical modified Yang-Baxter equation and connects it to Poisson-Lie groupoids, expanding understanding of symmetries in the WZNW phase space.

Findings

Derived explicit dynamical exchange r-matrices with Poisson-Lie symmetries.

Established the link between monodromy-dependent forms and exchange r-matrices.

Proposed a new interpretation of the dynamical Yang-Baxter equation in terms of groupoids.

Abstract

The precise relationship between the arbitrary monodromy dependent 2-form appearing in the chiral WZNW symplectic form and the `exchange r-matrix' that governs the corresponding Poisson brackets is established. Generalizing earlier results related to diagonal monodromy, the exchange r-matrices are shown to satisfy a new dynamical generalization of the classical modified Yang-Baxter equation, which is found to admit an interpretation in terms of (new) Poisson-Lie groupoids. Dynamical exchange r-matrices for which right multiplication yields a classical or a Poisson-Lie symmetry on the chiral WZNW phase space are presented explicitly.