Algebra of Dunkl Laplace-Runge-Lenz vector

We consider 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 generalizes the usual Laplace-Runge-Lenz vector and its components commute with Dunkl-Coulomb Hamiltonian given as Dunkl Laplacian with additional Coulomb potential $\gamma/r$. 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 $R_{g,0}(W)$ 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 algebras 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-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 Coulomb problem in $\mathbb R^N$. We also apply Dunkl Laplace-Runge-Lenz vector to establish maximal superintegrability of generalised Calogero-Moser systems.

💡 Research Summary

The paper introduces a Dunkl‑deformed Laplace‑Runge‑Lenz (LRL) vector for a finite Coxeter group W acting on ℝⁿ with a W‑invariant multiplicity function g. Starting from the Dunkl operators ∇_ξ, which are commuting differential‑difference operators containing reflection terms weighted by g, the authors construct a vector‑valued operator

L_i = ½ { x_i , ∇² } + γ x_i / r – Σ_{α∈R_+} g_α (α_i / (α·x)) s_α,

where ∇² = Σ_j ∇_j² is the Dunkl Laplacian, r = |x|, γ is the Coulomb coupling, and s_α are the reflections. They prove directly that each component commutes with the Dunkl‑Coulomb Hamiltonian

H_{g,γ} = ∇² + 2γ / r,

so the L_i are conserved quantities of the Dunkl‑Coulomb problem.

The main algebraic object of the work is the symmetry algebra R_{g,γ}(W) generated by the Dunkl LRL components {L_i}, the Dunkl angular momenta L_{ij}=x_i∇j−x_j∇i, and the group algebra ℂW, subject to quadratic relations that involve the auxiliary elements S{ij}=δ{ij}+2 Σ_{α∈R_+} g_α (α_iα_j/(α·α)) s_α. The authors establish that R_{g,γ}(W) possesses the Poincaré‑Birkhoff‑Witt (PBW) property: its associated graded algebra is a symmetric algebra, showing that the defining relations are a flat deformation of a polynomial algebra.

A central result is that a suitable central quotient of R_{g,γ}(W) is isomorphic to a central quotient of the Dunkl angular‑momentum algebra H_g^{so(N+1)}(W), which is generated by the same angular momenta together with the Coxeter group acting on the first N components of ℂ^{N+1}. This identifies H_g^{so(N+1)}(W) as the hidden symmetry algebra of the Dunkl‑Coulomb problem, extending the classical hidden so(N+1) symmetry of the ordinary Coulomb problem. When the multiplicity g vanishes, the isomorphism reduces to the well‑known identification of the hidden symmetry algebra with a quotient of U(so(N+1)).

In the special case γ=0 (no Coulomb term) the algebra R_{g,0}(W) embeds as a subalgebra of the rational Cherednik algebra H_g(W). Thus the Dunkl‑LRL construction provides a bridge between the Cherednik framework and the hidden symmetry viewpoint.

The authors then apply the algebraic structure to generalized Calogero‑Moser systems. By combining the Dunkl LRL vectors with the Dunkl angular momenta, they produce a family of mutually commuting operators that supplement the known integrals of motion. This yields maximal superintegrability for Calogero‑Moser models associated with any root system R, with or without an additional Coulomb potential. Moreover, they exhibit extra integrals for systems lacking full Coxeter symmetry, linking these to special representations of rational Cherednik algebras.

Overall, the paper achieves three significant advances: (1) a systematic Dunkl‑deformation of the LRL vector and proof of its conservation; (2) a detailed algebraic description of the resulting symmetry algebra, including PBW and quadratic central‑quotient properties; (3) a unified proof of maximal superintegrability for a broad class of quantum many‑body systems, thereby extending classical hidden symmetry concepts to the non‑local, reflection‑symmetric setting of Dunkl operators. Future directions suggested include representation theory of the Dunkl‑LRL algebra, extensions to non‑crystallographic groups, and physical applications in models where reflection symmetry and inverse‑square interactions play a role.