What is... the elliptic Calogero-Moser space?
Maxime Fairon (University of Glasgow)
Tuesday 10th December, 2019 16:00-17:00 Maths 311B
The Calogero-Moser (CM) system is an integrable system describing the classical motion of particles on a line which are subject to a potential of interaction that is either rational, trigonometric/hyperbolic or elliptic. With the exception of the elliptic case, it is well-known that the phase space of the CM system can be obtained by Hamiltonian reduction of a suitable space of matrices. It is then a natural question to ask if such a reduction picture also exists in the elliptic case if we start with a finite-dimensional space of matrices.
To tackle this problem in the complex setting, I will interpret the known phase spaces in terms of specific non-commutative algebras. Moreover, I will translate the Poisson structure of these phase spaces on the latter algebras using a version of non-commutative Poisson geometry introduced by Van den Bergh. I will then explain what is the natural elliptic analogue of these non-commutative algebras, and how this leads to the elliptic CM space. This is based on joint work with Oleg Chalykh (Leeds).