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

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 $B{C}_n$ 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 $B{C}_n$ 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.

shortened to 13 pages in v2 on request of Lett. Math. Phys. and corrected some spelling errors

References in corpus (3)

• Physics and Mathematics of Calogero particles
• Spin Calogero models associated with Riemannian symmetric spaces of negative curvature
• Notes on Collective Field Theory of Matrix and Spin Calogero Models

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• Quantum-classical duality for Gaudin magnets with boundary
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