Hidden algebra of the N-body Calogero problem

A certain generalization of the algebra gl(N,R) of first-order differential operators acting on a space of inhomogeneous polynomials in R is constructed. The generators of this (non)Lie algebra depend on permutation operators. It is shown that the Hamiltonian of the N -body Calogero model can be represented as a second-order polynomial in the generators of this algebra. Given representation implies that the Calogero Hamiltonian possesses infinitely-many finite-dimensional invariant subspaces with explicit bases, which are closely related to the finite-dimensional representations of above algebra. This representation is an alternative to the standard representation of the Bargmann-Fock type in terms of creation and annihilation operators. On leave of absence from the Institute for Theoretical and Experimental Physics, Moscow 117259, Russia E-mail: turbiner@cernvm or [email protected] Supported in part by a CAST grant of the US National Academy of Sciences. The Calogero model [1] is a quantum mechanical system of N particles on a line interacting via a pairwise potential and defined by the Hamiltonian HCal = 1 2 N