Speaker: D. Talalaev (MSU,YarSU)

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

Date and time:  23.04.2025, 17:00 (GMT +03:00)

Abstract:  The talk is focused on the 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. I will describe the application of the bi-Hamiltonian formalism for some pencils from this family, namely a method for constructing involutive subalgebras for a linear bracket starting by the center of the quadratic bracket. I will provide some interesting examples of families of this type.

An important ingredient of the construction is the family of antidiagonal principal minors of the Lax matrix. A crucial but quite unbiguous condition of the log-canonicity of  brackets of these minors with all the generators of the Poisson algebra establishes a relation of our families with cluster algebras, a similar property arises in the context of Poisson structures consistent with mutations.

The talk is based on the recent joint paper with V.V. Sokolov https://arxiv.org/abs/2502.16925

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com