$C_{\lambda}$-extended oscillator algebras and some of their deformations and applications to quantum mechanics

Abstract

$C_{\lambda}$-extended oscillator algebras generalizing the Calogero-Vasiliev algebra, where $C_{\lambda}$ is the cyclic group of order $\lambda$, are studied both from mathematical and applied viewpoints. Casimir operators of the algebras are obtained, and used to provide a complete classification of their unitary irreducible representations under the assumption that the number operator spectrum is nondegenerate. Deformed algebras admitting Casimir operators analogous to those of their undeformed counterparts are looked for, yielding three new algebraic structures. One of them includes the Brzezi\'nski {\em et al.} deformation of the Calogero-Vasiliev algebra as a special case. In its bosonic Fock-space representation, the realization of $C_{\lambda}$-extended oscillator algebras as generalized deformed oscillator ones is shown to provide a bosonization of several variants of supersymmetric quantum mechanics: parasupersymmetric quantum mechanics of order $p = \lambda-1$ for any $\lambda$, as well as pseudosupersymmetric and orthosupersymmetric quantum mechanics of order two for $\lambda=3$.