Деформационное квантование и квантовые группы. Семинар 2, Г.И.Шарыгин, А.Б.Жеглов

29.09.26 Докладчик: Dmitriy Timashev (Lomonosov MSU) Maximal commutative Poisson subalgebras, bi-Poisson structures, and 2-decompositions of Lie algebras Максимальные коммутативные подалгебры Пуассона, бипуассоновы структуры и 2-разложения алгебр Ли Abstract: In the theory of integrable systems, one of the fundamental problems is to construct a complete set of first integrals in involution for a Hamiltonian dynamical system. In algebraic terms, the problem consists of constructing a commutative subalgebra of maximum transcendence degree in a given Poisson algebra F. One of the methods for constructing commutative Poisson subalgebras is the Lenard–Magri scheme, which is based on the use of a bi‑Poisson structure, i.e., another Poisson bracket on F, consistent with the original one. This scheme, in particular, encompasses the well‑known Mishchenko–Fomenko argument shift method. In the case where F = S(g) is the symmetric algebra of the Lie algebra g with the Poisson–Lie bracket, and the second Poisson bracket is also linear, i.e., it defines another Lie algebra structure on g, a sheaf of consistent Lie brackets on g arises, parametrized (up to proportionality) by points of the projective line. We will show that under certain assumptions, for example, if the Lie algebra g is semisimple, the Lenard–Magri scheme leads in this situation to a commutative Poisson subalgebra of maximum transcendence degree if and only if all parameter values in the pencil are regular, i.e., the indices of the Lie algebras corresponding to different parameter values coincide. As an application, we will consider two constructions of a pencil of consistent Lie brackets. One of them, proposed by Panyushev and Yakimova (2021), is based on a 2-decomposition of the Lie algebra g, i.e., a decomposition g = f + h into the direct sum of two Lie subalgebras. In this case, we will show that the commutative Poisson subalgebra constructed according to the Lenard–Magri scheme has the maximum degree of transcendence if and only if the subalgebras f and h are spherical (in a special case, this is a result of Panyusheva and Yakimova). The second construction for g = gl(n) uses the A-bracket: Lie [X,Y]_A = XAY - YAX, consistent with the standard matrix commutator (here A is a fixed matrix from gl(n)). It turns out that the corresponding commutative Poisson subalgebra C has the maximum degree of transcendence if and only if the matrix A is regular, i.e., the geometric multiplicities of all its eigenvalues are equal to 1. In this case, C coincides with the Mishchenko–Fomenko algebra constructed by shifting the argument along -A. These results belong to D.A. Astrelina (2026). Лекторы - Георгий Игоревич Шарыгин, Александр Борисович Жеглов Страница курса - https://mccme.ru/ru/nmu/courses-of-nmu/osen-20262027/nmu_autumn2026_deformacionnoe-kvantovanie/ Плейлист на YouTube - https://www.youtube.com/playlist?list=PLJW37wfUitSk Плейлист на RuTube - https://rutube.ru/plst/1785006 Канал НМУ на RuTube - https://rutube.ru/channel/42881756/