On the coincidence of two manifolds associated with the Calogero model

Abstract

Let R be a root system, for example, the root system associated to a semisimple Lie algebra. In [1], V. A. Golubeva and V. P. Lexin constructed two algebraic manifolds (Bethe and Dunkl manifolds) using the "universal" Dunkl operators. These manifolds were defined as subsets of the complex space ? N of dimension equal to the number of roots of the root system under consideration. The first manifold (Bethe manifold) is characterized by the following property: the Laplace operator constructed by means of Dunkl operators coincides with the "universal" Hamiltonian of the Calogero model. The second one (Dunkl manifold) is characterized by the property: the "universal" Dunkl operators commute. In this paper, the manifolds associated with the irreducible root system of Coxeter type are considered. We give their construction supposing that these manifolds are embedded in ? N/2. A theoremon the coincidence of Bethe and Dunkl manifolds is proved.