Algebra of Dunkl Laplace-Runge-Lenz vector
We introduce Dunkl version of Laplace-Runge-Lenz vector associated with a finite Coxeter group $W$ acting geometrically in $\mathbb R^N$ with multiplicity function $g$. This vector commutes with Dunkl Laplacian with additional Coulomb potential $\gamma/r$, and it generalises the usual Laplace-Runge-Lenz vector. We study resulting symmetry algebra $R_{g, \gamma}(W)$ and show that it has Poincar\'e-Birkhoff-Witt property. In the absence of Coulomb potential this symmetry algebra is a subalgebra of the rational Cherednik algebra $H_g(W)\supset R_{g,0}(W)$. We show that its central quotient is a quadratic algebra isomorphic to a central quotient of the corresponding Dunkl angular momenta algebra $H_g^{so(N+1)}(W)$. This gives interpretation of the algebra $H_g^{so(N+1)}(W)$ as the hidden symmetry algebra of Dunkl Laplacian. On the other hand 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 Coulomb problem in ${\mathbb R}^N$.
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07/16/19 06:04PM
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