HIDDEN ALGEBRAS OF THE (SUPER) CALOGERO AND SUTHERLAND MODELS

We propose to parametrize the configuration space of one-dimensional q uantum systems of N identical particles by the elementary symmetric po lynomials of bosonic and fermionic coordinates. It is shown that in th is parametrization the Hamiltonians of the A(N), BCN, B-N, C-N and D-N Calogero and Sutherland models, as well as their supersymmetric gener alizations, can be expressed-for arbitrary values of the coupling cons tants-as quadratic polynomials in the generators of a Borel subalgebra of the Lie algebra gl(N + 1) or the Lie superalgebra gl(N + 1 N) for the supersymmetric case. These algebras are realized by first order di fferential operators. This fact establishes the exact solvability of t he models according to the general definition given by Turbiner, and i mplies that the Calogero and Jack-Sutherland polynomials, as well as t heir supersymmetric generalizations, are related to finite-dimensional irreducible representations of the Lie algebra gl(N + 1) and the Lie superalgebra gl(N + 1 N). (C) 1998 American Institute of Physics.