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      Algebra of Dunkl Laplace-Runge-Lenz vector

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          Abstract

          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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          Poincaré–Birkhoff–Witt Theorem for Quadratic Algebras of Koszul Type

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            A Remark on the Dunkl Differential—Difference Operators

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              Exchange operator formalism for integrable systems of particles

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                Author and article information

                Journal
                15 July 2019
                Article
                1907.06706
                0e7b6ac5-f731-4b46-b1ed-9ff6cdc7982f

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                23 pages
                math-ph hep-th math.MP math.QA nlin.SI

                Mathematical physics,High energy & Particle physics,Mathematical & Computational physics,Nonlinear & Complex systems,Algebra

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