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.