Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1997-05-28
J.Math.Phys. 39 (1998) 1285-1315
Physics
High Energy Physics
High Energy Physics - Theory
35 pages, plain LaTeX. Paper expanded, some formulas for SUSY hamiltonians corrected. Appendix with clarifying technical detai
Scientific paper
10.1063/1.532380
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).
Brink Lars
Turbiner Alexander
Wyllard Niclas
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