A Quadratic Deformation of the Heisenberg-Weyl and Quantum Oscillator Enveloping Algebras

TL;DR

This paper introduces a new quadratic deformation of the quantum oscillator algebra and explores its representations, symmetries, and connections to other deformed algebras, revealing novel algebraic structures at roots of unity.

Contribution

It presents a novel 2-parameter quadratic deformation of the quantum oscillator algebra and analyzes its representations and symmetries, including connections to deformed Virasoro and $SL_q(2)$ algebras.

Findings

Finite dimensional representations at roots of unity

Construction of $SL_q(2)$ symmetry

Deformation of Virasoro algebra at roots of unity

Abstract

A new 2-parameter quadratic deformation of the quantum oscillator algebra and its 1-parameter deformed Heisenberg subalgebra are considered. An infinite dimensional Fock module representation is presented which at roots of unity contains null vectors and so is reducible to a finite dimensional representation. The cyclic, nilpotent and unitary representations are discussed. Witten's deformation of $sl_2$ and some deformed infinite dimensional algebras are constructed from the $1d$ Heisenberg algebra generators. The deformation of the centreless Virasoro algebra at roots of unity is mentioned. Finally the $SL_q(2)$ symmetry of the deformed Heisenberg algebra is explicitly constructed.