On interpretations and constructions of classical dynamical r-matrices

TL;DR

This paper provides a finite-dimensional description of the relationship between monodromy-dependent forms and exchange r-matrices in the chiral WZNW model, including the canonical solution of the classical dynamical Yang-Baxter equation.

Contribution

It offers a new finite-dimensional perspective on exchange r-matrices and their connection to the symplectic form in the chiral WZNW model, especially for the canonical case.

Findings

Established a direct link between monodromy forms and exchange r-matrices.

Developed the special case of the canonical solution on self-dual Lie algebras.

Clarified the structure of exchange r-matrices in the classical dynamical Yang-Baxter context.

Abstract

In this note we complement recent results on the exchange $r$-matrices appearing in the chiral WZNW model by providing a direct, purely finite-dimensional description of the relationship between the monodromy dependent 2-form that enters the chiral WZNW symplectic form and the exchange $r$-matrix that governs the corresponding Poisson brackets. We also develop the special case in which the exchange $r$-matrix becomes the `canonical' solution of the classical dynamical Yang-Baxter equation on an arbitrary self-dual Lie algebra.