Seminars: D. V. Talalaev, On a family of Poisson brackets on $\mathfrak{gl}(n)$ compatible with the Sklyanin bracket

SEMINARS


Seminar of the Department of Theoretical Physics, Steklov Mathematical Institute of RAS February 4, 2026 14:00, Moscow, Steklov Mathematical Institute of RAS, Room 313 (8 Gubkina)

On a family of Poisson brackets on $\mathfrak{gl}(n)$ compatible with the Sklyanin bracket D. V. Talalaev^abc ^a State Scientific Center of the Russian Federation - Institute for Theoretical and Experimental Physics, Moscow ^b Lomonosov Moscow State University ^c P.G. Demidov Yaroslavl State University
Abstract: I will tell about a recent result obtained jointly with V. V. Sokolov in arXiv:2502.16925. We have constructed a family of compatible quadratic Poisson brackets on $\mathfrak{gl}(n)$ generalizing the Sklyanin bracket. The question itself is closely related to the inverse scattering problem in the theory of integrable systems, the formalism of Lie-bialgebras and Poisson-Lie groups. The resulting family of quadratic brackets is directly related to the algebras of the reflection equation type and is a generalization of brackets of the Sklyanin type. We develop the argument shift method in the context of the resulting family of quadratic brackets and associated linear ones. Using the Bi-Hamiltonian formalism we construct several interesting families of involutive subalgebras relevant both to the theory of integrable systems and to the general problem of invariants. Another phenomenon of the proposed construction consists in a special condition for the brackets of the antidiagonal minors of the Lax matrix for the entire family of quadratic brackets under consideration: these brackets have a log-canonical form. This property is related to canonical Poisson structures on cluster algebras and is significantly used to construct involutive subalgebras.