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We study a holomorphic Poisson structure defined on the linear space$S(n,d):= {\rm Mat}_{n\times d}(\mathbb{C}) \times {\rm Mat}_{d\timesn}(\mathbb{C})$ that is covariant under the natural left actions of thestandard ${\rm GL}(n,\mathbb{C})$ and ${\rm GL}(d,\mathbb{C})$ Poisson-Liegroups. The Poisson brackets of the matrix elements contain quadratic andconstant 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 weconstruct a local Poisson map from the Cartesian product of $d$ independentcopies of $S(n,1)$ into $S(n,d)$, which is a holomorphic diffeomorphism in aneighborhood of zero. The Poisson structure on $S(n,d)$ is the complexificationof a real Poisson structure on ${\rm Mat}_{n\times d}(\mathbb{C})$ constructedby the authors and Marshall, where a similar decoupling into $d$ independentcopies 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 complextrigonometric spin Ruijsenaars-Schneider system by Hamiltonian reduction.

journal of mathematical physics

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