A class of Calogero type reductions of free motion on a simple Lie group

TL;DR

This paper explores reductions of free motion on simple Lie groups leading to new spin Calogero models and standard $BC_n$ Sutherland models, extending classical derivations and allowing for quantization.

Contribution

It introduces a new class of Calogero type models from Lie group reductions, generalizing Olshanetsky-Perelomov's derivation and enabling quantized versions.

Findings

Derivation of new spin Calogero models from Lie group reductions.

Recovery of the $BC_n$ Sutherland model at special parameter values.

Extension of the reduction framework to quantized systems.

Abstract

The reductions of the free geodesic motion on a non-compact simple Lie group G based on the $G_+ \times G_+$ symmetry given by left- and right multiplications for a maximal compact subgroup $G_+ \subset G$ are investigated. At generic values of the momentum map this leads to (new) spin Calogero type models. At some special values the `spin' degrees of freedom are absent and we obtain the standard $BC_n$ Sutherland model with three independent coupling constants from SU(n+1,n) and from SU(n,n). This generalization of the Olshanetsky-Perelomov derivation of the $BC_n$ model with two independent coupling constants from the geodesics on $G/G_+$ with G=SU(n+1,n) relies on fixing the right-handed momentum to a non-zero character of $G_+$. The reductions considered permit further generalizations and work at the quantized level, too, for non-compact as well as for compact G.