On a family of Poisson brackets on gl(n) compatible with the Sklyanin bracket

TL;DR

This paper explores a family of quadratic Poisson brackets on gl(n) that extend the Sklyanin bracket, utilizing bi-Hamiltonian formalism to construct involutive subalgebras and analyze their centers via principal minors.

Contribution

It introduces a new family of compatible quadratic Poisson brackets on gl(n), generalizing Sklyanin brackets and applying bi-Hamiltonian methods for algebraic construction.

Findings

Constructed centers of quadratic brackets using principal minors.

Identified conditions for log-canonicity of minors with generators.

Demonstrated applications to bi-Hamiltonian formalism.

Abstract

In this paper, we study a family of compatible quadratic Poisson brackets on gl(n), generalizing the Sklyanin one. For any of the brackets in the family, the argument shift determines the compatible linear bracket. The main interest for us is the use of the bi-Hamiltonian formalism for some pencils from this family, as a method for constructing involutive subalgebras for a linear bracket starting by the center of the quadratic bracket. We give some interesting examples of families of this type. We construct the centers of the corresponding quadratic brackets using the antidiagonal principal minors of the Lax matrix. Special attention should be paid to the condition of the log-canonicity of the brackets of these minors with all the generators of the Poisson algebra of the family under consideration. A similar property arises in the context of Poisson structures consistent with cluster transformations.