Seminar talk, 6 November 2024

Speaker: Georgy Sharygin

Title: Remarkable properties of the full symmetric Toda system

Abstract:
A full symmetric Toda system is a Hamiltonian dynamical system on the space of symmetric real matrices with zero trace, generalizing the usual open Toda chain. This system is given by the Lax equation ${\dot {L}}=[L,M(L)]$, where $M(L)$ is the (naive) antisymmetrization of the symmetric matrix $L$: the difference of its super and subdiagonal parts (with zeros on the diagonal). The Hamiltonianity of this system comes from the identification of the space of symmetric matrices with the space dual to the algebra of upper triangular matrices, with the Hamilton function being $1/2Tr(L^{2})$. This system can be further generalized to obtain systems on the spaces of "generalized symmetric matrices", the symmetric components of the Cartan expansion of the semi-simple real Lie algebras. In a somewhat unexpected way, all these systems turn out to be integrable (in the sense of having a sufficiently large commutative algebra of first integrals) and possess a number of remarkable properties which I will discuss: their trajectories always connect fixed points corresponding to the elements of the Weyl group of the original Lie algebra, and two such points are connected if and only if the elements of the Weyl group are comparable in Bruhat order; in the case of a system on spaces of generalized symmetric matrices, this property allows one to describe the intersections of the real Bruhat cells; this system has a large set of symmetries (sufficient for it to be Lie-Bianchi integrable); its additional first integrals can be obtained by a "cut" procedure, and the trajectories of the corresponding Hamiltonian fields can be obtained by the QR decomposition; if time permits, I will describe alternative families of first integrals (commutative and non-commutative); finally, I will describe a way to lift the extra first integrals of the "cut" into the universal enveloping algebra with commutativity preserved.

The talk is based on a series of works by the author jointly with Yu. Chernyakov, A. Sorin and D. Talalaev.

Video