A decoupling property of some Poisson structures on Matn×d(C)×Matd×n(C) supporting GL(n,C)×GL(d,C) Poisson–Lie symmetry

Abstract

We study a holomorphic Poisson structure defined on the linear space S(n,d)≔Matn×d(C)×Matd×n(C) that is covariant under the natural left actions of the standard GL(n,C) and GL(d,C) Poisson–Lie groups. The Poisson brackets of the matrix elements contain quadratic and constant terms, and the Poisson tensor is non-degenerate on a dense subset. Taking the d = 1 special case gives a Poisson structure on S(n, 1), and we construct a local Poisson map from the Cartesian product of d independent copies of S(n, 1) into S(n, d), which is a holomorphic diffeomorphism in a neighborhood of 0. The Poisson structure on S(n, d) is the complexification of a real Poisson structure on Matn×d(C) constructed by the authors and Marshall, where a similar decoupling into d independent copies was observed. We also relate our construction to a Poisson structure on S(n, d) defined by Arutyunov and Olivucci in the treatment of the complex trigonometric spin Ruijsenaars–Schneider system by Hamiltonian reduction.