Algebra of Dunkl Laplace–Runge–Lenz vector

Abstract

We introduce the Dunkl version of the Laplace–Runge–Lenz vector associated with a finite Coxeter group W acting geometrically in mathbb R^N and with a multiplicity function g. This vector generalizes the usual Laplace–Runge–Lenz vector and its components commute with the Dunkl–Coulomb Hamiltonian given as the Dunkl Laplacian with an additional Coulomb potential gamma /r. We study the resulting symmetry algebra R_{g, gamma }(W) and show that it has the Poincaré–Birkhoff–Witt property. In the absence of a Coulomb potential, this symmetry algebra R_{g,0}(W) is a subalgebra of the rational Cherednik algebra H_g(W). We show that a central quotient of the algebra R_{g, gamma }(W) is a quadratic algebra isomorphic to a central quotient of the corresponding Dunkl angular momenta algebra H_g^{so(N+1)}(W). This gives an interpretation of the algebra H_g^{so(N+1)}(W) as the hidden symmetry algebra of the Dunkl–Coulomb problem in mathbb {R}^N. By specialising R_{g, gamma }(W) to g=0, we recover a quotient of the universal enveloping algebra U(so(N+1)) as the hidden symmetry algebra of the Coulomb problem in {mathbb R}^N. We also apply the Dunkl Laplace–Runge–Lenz vector to establish the maximal superintegrability of the generalised Calogero–Moser systems.