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

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 $s{l}_2$ 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 $S{L}_q(2)$ symmetry of the deformed Heisenberg algebra is explicitly constructed.

23 pages of plain TeX (with phyzzx.tex macros). I've made a few minor corrections and added 2 references