Algebra of Dunkl Laplace–Runge–Lenz vector

Abstract

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