On Dunkl angular momenta algebra

TL;DR

This paper explores the algebraic structure of Dunkl angular momentum generators within the rational Cherednik algebra, revealing quadratic relations, PBW properties, and connections to Calogero-Moser Hamiltonians, with extensions to gl(N).

Contribution

It characterizes the Dunkl angular momenta algebra, establishes its quadratic PBW structure, and links it to integrable Calogero-Moser systems, including a gl(N) generalization.

Findings

The algebra has quadratic defining relations.

It contains the angular part of the Calogero-Moser Hamiltonian.

The algebra is of PBW type and includes the Calogero-Moser Hamiltonian as a central element.

Abstract

We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincare-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl(N) version of the subalgebra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.