Superintegrable Systems on Moduli Spaces of Flat Connections

The main result of this paper is the construction of a family of superintegrable Hamiltonian systems on moduli spaces of flat connections on a principal G-bundle on a surface. The moduli space is a Poisson variety with Atiyah–Bott Poisson structure. Among particular cases of such systems are spin generalizations of Ruijsenaars–Schneider models.

Superintegrability of Generalized Toda Models on Symmetric Spaces

In this paper we prove superintegrability of Hamiltonian systems generated by functions on $K\backslash G/K$, restricted to a symplectic leaf of the Poisson variety $G/K$, where $G$ is a simple Lie group with the standard Poisson Lie structure, and $K$ is its subgroup of fixed points with respect to the Cartan involution.

ON THE DEFORMATION QUANTIZATION OF AFFINE ALGEBRAIC VARIETIES

We compute an explicit algebraic deformation quantization for an affine Poisson variety described by an ideal in a polynomial ring, and inheriting its Poisson structure from the ambient space.