The Dunkl two-photon/Schrödinger algebras and their applications
Abstract
The work introduces the novel Dunkl two-photon and Dunkl-Schrödinger algebras as reflection-extended counterparts of the standard six-dimensional two-photon and Schrödinger algebras. These extensions are realized through the presence of an involutive generator representing a reflection, and they reduce to the undeformed two-photon and Schrödinger algebras in the absence of this additional generator in an appropriate limit. For both algebraic structures, which are shown to be isomorphic, we derive the associated cubic polynomial Casimir invariant. Furthermore, we identify several structurally significant subalgebras and compute their corresponding quadratic Casimir invariants. Representations of the Dunkl two-photon algebra are constructed on the Fock space of the one-dimensional Dunkl harmonic oscillator and subsequently translated into holomorphic realizations within the associated Dunkl-Fock-Bargmann representation. Finally, for the Dunkl-Schrödinger algebra, we present an explicit vector field realization in terms of the Dunkl spatial derivative, while keeping time continuous. When applied to the Casimir invariants of the aforementioned subalgebras, this realization yields a family of six nontrivial Dunkl differential-difference equations in 1+1 dimensions, including, in particular, a Dunkl version of the well-known heat-Schrödinger equation.
Related
- Lp estimates for Dunkl--Schrödinger operators
- Riesz transform characterization of Hardy spaces associated with Dunkl--Schrödinger operators
- On Lipschitz Spaces, Semigroup Transference, and Heat Kernel Derivative Estimates in Dunkl--Schrödinger Analysis
- Dunkl theory, convolution algebras, and related Markov processes
- Matching algebras and their Dunkl subalgebras
BibTeX
@misc{herranz2026dunkl,
title = {{The Dunkl two-photon/Schrödinger algebras and their applications}},
author = {Francisco J. Herranz and Danilo Latini},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.31104},
eprint = {2608.31104},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.31104},
}
RIS
TY - UNPB
TI - The Dunkl two-photon/Schrödinger algebras and their applications
AU - Francisco J. Herranz
AU - Danilo Latini
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.31104
UR - https://arxiv.org/abs/2608.31104
AB - The work introduces the novel Dunkl two-photon and Dunkl-Schrödinger algebras as reflection-extended counterparts of the standard six-dimensional two-photon and Schrödinger algebras. These extensions are realized through the presence of an involutive generator representing a reflection, and they reduce to the undeformed two-photon and Schrödinger algebras in the absence of this additional generator in an appropriate limit. For both algebraic structures, which are shown to be isomorphic, we derive the associated cubic polynomial Casimir invariant. Furthermore, we identify several structurally significant subalgebras and compute their corresponding quadratic Casimir invariants. Representations of the Dunkl two-photon algebra are constructed on the Fock space of the one-dimensional Dunkl harmonic oscillator and subsequently translated into holomorphic realizations within the associated Dunkl-Fock-Bargmann representation. Finally, for the Dunkl-Schrödinger algebra, we present an explicit vector field realization in terms of the Dunkl spatial derivative, while keeping time continuous. When applied to the Casimir invariants of the aforementioned subalgebras, this realization yields a family of six nontrivial Dunkl differential-difference equations in 1+1 dimensions, including, in particular, a Dunkl version of the well-known heat-Schrödinger equation.
ER -