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

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 $Ł\subset \A$ to those on another pair $\K\subset \A$, where $\K\subset Ł\subset \A$ is a chain of Lie algebras for which $Ł$ admits a reductive decomposition as $Ł=\K+\M$. 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.