The classical r-matrix in a geometric framework

Abstract

We use a Riemannian (or pseudo-Riemannian) geometric framework to formulate the theory of the classical r-matrix for integrable systems. In this picture the r-matrix is related to a fourth rank tensor, named the r-tensor, on the configuration space. The r-matrix itself carries one connection type index and three tensorial indices. Being defined on the configuration space it has no momentum dependence but is dynamical in the sense of depending on the configuration variables. The tensorial nature of the r-matrix is used to derive its transformation properties. The resulting transformation formula turns out to be valid for a general r-matrix structure independently of the geometric framework. Moreover, the entire structure of the r-matrix equation follows directly from a simple covariant expression involving the Lax matrix and its covariant derivative. Therefore it is argued that the geometric formulation proposed here helps to improve the understanding of general r-matrix structures. It is also shown how the Jacobi identity gives rise to a generalized dynamical classical Yang-Baxter equation involving the Riemannian curvature.