Chiral Extensions of the WZNW Phase Space, Poisson-Lie Symmetries and Groupoids

TL;DR

This paper explores the mathematical structure of the chiral WZNW phase space, establishing a link between monodromy-dependent forms, exchange r-matrices, and Poisson-Lie groupoids, revealing new dynamical Yang-Baxter equations.

Contribution

It introduces a new dynamical generalization of the classical modified Yang-Baxter equation and constructs Poisson-Lie groupoids encoding this relation for arbitrary simple Lie groups.

Findings

Derived the inverse of the chiral WZNW symplectic form.

Established a relationship between monodromy forms and exchange r-matrices.

Constructed Poisson-Lie groupoids corresponding to the generalized Yang-Baxter equation.

Abstract

The chiral WZNW symplectic form $\Omega^{\rho}_{chir}$ is inverted in the general case. Thereby a precise relationship between the arbitrary monodromy dependent 2-form appearing in $\Omega^{\rho}_{chir}$ and the exchange r-matrix that governs the Poisson brackets of the group valued chiral fields is established. The exchange r-matrices are shown to satisfy a new dynamical generalization of the classical modified Yang-Baxter (YB) equation and Poisson-Lie (PL) groupoids are constructed that encode this equation analogously as PL groups encode the classical YB equation. For an arbitrary simple Lie group G, exchange r-matrices are found that are in one-to-one correspondence with the possible PL structures on G and admit them as PL symmetries.