Calogero-Moser systems with periodic and doubly periodic interaction potentials and loop algebras (Q2774697)

For a finite number of particles on the line pairwise interacting with \(1/r^2\) potential, the positions are given by the eigenvalues of some time-dependent matrix. Infinite periodic or doubly periodic replication of the particles yields Calogero-Moser systems with periodic or doubly periodic interaction potential. We are thus led to consider matrices of infinite order, which are identified with Fourier series with matrix coefficients, depending on an additional parameter. These distributional loops of matrices (tori of matrices in the doubly periodic case) are shown to obey simple (partial) differential equations, which allow us to determine them explicitly. Thus we obtain the already known solution of the Calogero-Moser system on the circle, and provide a new insight for the system on the torus.