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- 1999 (2) (remove)

The Hamiltonian of the \(N\)-particle Calogero model can be expressed in terms of generators of a Lie algebra for a definite class of representations. Maintaining this Lie algebra, its representations, and the flatness of the Riemannian metric belonging to the second order differential operator, the set of all possible quadratic Lie algebra forms is investigated. For \(N = 3\) and \(N = 4\) such forms are constructed explicitly and shown to correspond to exactly solvable Sutherland models. The results can be carried over easily to all \(N\).

Trigonometric invariants are defined for each Weyl group orbit on the root lattice. They are real and periodic on the coroot lattice. Their polynomial algebra is spanned by a basis which is calculated by means of an algorithm. The invariants of the basis can be used as coordinates in any cell of the coroot space and lead to an exactly solvable model of Sutherland type. We apply this construction to the \(F_4\) case.