On a family of Poisson brackets on gl(n) compatible with the Sklyanin bracket
Vladimir V. Sokolov, Dmitry V. Talalaev
TL;DR
The paper develops a framework of compatible quadratic Poisson brackets on $\mathfrak{gl}_n$ that generalize the Sklyanin bracket and constructs a compatible linear pencil via the argument shift. It provides an explicit four-parameter family and analyzes the centers of both the quadratic and linear brackets in terms of antidiagonal minors of the Lax matrix, establishing log-canonical relations and bi-Hamiltonian reduction pathways. A detailed study of a 4-parameter subfamily identifies canonical specializations including Sklyanin, rho-type, and Toda-relevant brackets, and demonstrates how centers generate commutative subalgebras across the pencil. The results illuminate structural links to RTT/RE algebras, Poisson reductions to Toda chains, and potential quantization and cluster-structure interpretations, with explicit central elements described for several parameter choices.
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.