A decoupling property of some Poisson structures on ${\mathrm{Mat}}_{n\times d}(\mathbb{C})\times {\mathrm{Mat}}_{d\times n}(\mathbb{C})$ supporting $\mathrm{GL}(n,\mathbb{C})\times \mathrm{GL}(d,\mathbb{C})$ Poisson-Lie symmetry

Abstract

We study a holomorphic Poisson structure defined on the linear space $S(n,d):={\mathrm{Mat}}_{n\times d}(\mathbb{C})\times {\mathrm{Mat}}_{d\times n}(\mathbb{C})$ that is covariant under the natural left actions of the standard $\mathrm{GL}(n,\mathbb{C})$ and $\mathrm{GL}(d,\mathbb{C})$ 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 zero. The Poisson structure on $S(n,d)$ is the complexification of a real Poisson structure on ${\mathrm{Mat}}_{n\times d}(\mathbb{C})$ 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.

Accepted version

References in corpus (1)

• Trigonometric real form of the spin RS model of Krichever and Zabrodin

Cited by in corpus (1)

• Lax equations for relativistic $\mathrm{GL}(NM,\mathbb{C})$ Gaudin models on elliptic curve