Abstract:
The simplest way to define the full symmetric Toda system is to say that this is a system of Lax type:
$$
\dot L=[M(L),L],
$$
where the variable matrix $L$ is a real symmetric matrix of size $n\times n$, and $M(L)=L_+-L_-$ is its "naive" anti-symmetrization (the matrix composed of the upper triangular part $L_+$ of $L$ with the same sign and its lower triangular parts of $L_-$ with reversed sign). This system has a lot of interesting properties: it is a Hamiltonian system that is Liouville integrable, it is also noncommutatively integrable (in the sense of Nekhoroshev), its singular points and trajectories are ordered in accordance with the Bruhat order on the permutation group. Its generalizations to other semisimple Lie groups have similar properties. In my talk, I will give an outline of the proofs of some of these statements and tell you how to construct the symmetries of such a system. In particular, it will follow from this construction that Toda system is integrable in the sense of the Lie-Bianchi theorem (that is, it has a solvable algebra of symmetries of maximum dimension). The report is based on a series of joint papers with Yu. Chernyakov, D. Talalaev and A. Sorin.
