On dynamical r-matrices obtained from Dirac reduction and their generalizations to affine Lie algebras

TL;DR

This paper explores how Dirac reductions of Poisson manifolds lead to new dynamical r-matrices, including those associated with affine Lie algebras, expanding the known solutions to the classical dynamical Yang-Baxter equation.

Contribution

It introduces a method to generate new dynamical r-matrices via Dirac reduction and extends these constructions to affine Lie algebras, revealing new solutions.

Findings

Dirac reductions produce mappings between dynamical r-matrices.

Several known r-matrices are naturally recovered in this framework.

New r-matrices are constructed for affine Lie algebras, with the dynamical variable in the grade zero subalgebra.

Abstract

According to Etingof and Varchenko, the classical dynamical Yang-Baxter equation is a guarantee for the consistency of the Poisson bracket on certain Poisson-Lie groupoids. Here it is noticed that Dirac reductions of these Poisson manifolds give rise to a mapping from dynamical r-matrices on a pair $\L\subset \A$ to those on another pair $\K\subset \A$, where $\K\subset \L\subset \A$ is a chain of Lie algebras for which $\L$ admits a reductive decomposition as $\L=\K+\M$. Several known dynamical r-matrices appear naturally in this setting, and its application provides new r-matrices, too. In particular, we exhibit a family of r-matrices for which the dynamical variable lies in the grade zero subalgebra of an extended affine Lie algebra obtained from a twisted loop algebra based on an arbitrary finite dimensional self-dual Lie algebra.