Hidden Algebras of the (super) Calogero and Sutherland models

TL;DR

This paper demonstrates that various Calogero and Sutherland models, including their supersymmetric versions, can be expressed as quadratic polynomials in Lie algebra generators, establishing their exact solvability and linking their polynomials to Lie algebra representations.

Contribution

It introduces a parametrization of the configuration space using elementary symmetric polynomials, enabling Lie algebraic expressions of the models for all coupling constants.

Findings

Models are expressed as quadratic polynomials in Lie algebra generators.

Exact solvability is established via Lie algebraic realization.

Polynomials correspond to finite-dimensional Lie algebra representations.

Abstract

We propose to parametrize the configuration space of one-dimensional quantum systems of N identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates. It is shown that in this parametrization the Hamiltonians of the A_N, BC_N, B_N, C_N and D_N Calogero and Sutherland models, as well as their supersymmetric generalizations, can be expressed -- for arbitrary values of the coupling constants -- 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 differential operators. This fact establishes the exact solvability of the models according to the general definition given by one of the authors in 1994, and implies that the Calogero and Jack-Sutherland polynomials, as well as their supersymmetric generalizations, are related to finite-dimensional irreducible representations of the Lie algebra gl(N+1) and the Lie superalgebra gl(N+1|N).