Speaker: Dmitry Timashev (MSU)

Title: Maximal Poisson commutative subalgebras and 2-splittings of semisimple Lie algebras

Date and time:  1.10.2025, 17:00 (GMT +03:00)

Abstract: A fundamental problem in the theory of integrable systems is to construct a complete involutive set of first integrals for a Hamiltonian dynamical system. Algebraically the problem amounts to construction of a commutative subalgebra of maximal transcendence degree in a given Poisson algebra. A powerful method of constructing Poisson commutative subalgebras in the symmetric algebra S(g) of a semisimple Lie algebra g, known as the Lenard-Magri scheme, is based on including the Lie-Poisson bracket on S(g) in a pencil of compatible Poisson brackets. If all Poisson brackets in the pencil are linear, i.e., come from a pencil of Lie brackets on g, then the Lenard-Magri scheme yields a Poisson commutative subalgebra of maximal transcendence degree if and only if all values of the parameter in the pencil are regular, i.e. the indices of the Lie algebras corresponding to different values of the parameter are one and the same. Panyushev and Yakimova (2021) suggested an implementation of the Lenard-Magri scheme by considering a pencil of Lie brackets associated to a 2-splitting of g, i.e., a decomposition into a direct sum of Lie subalgebras g = f + h. The possible singular values of the parameter correspond to the Inönü-Wigner contractions of g along f and h. Extending the results of Panyushev and Yakimova, we prove a formula for the index of an Inönü-Wigner contraction of g and deduce that the resulting Poisson commutative subalgebra of S(g) has maximal transcendence degree if and only if both f and h are spherical Lie subalgebras.

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