C$_{\lambda}$-extended Oscillator Algebras: Theory and Applications to (Variants) of Supersymmetric Quantum Mechanics

Abstract

C$_{\lambda}$-extended oscillator algebras, where C$_{\lambda}$ is the cyclic group of order $\lambda$, are introduced and realized as generalized deformed oscillator algebras. For $\lambda=2$, they reduce to the well-known Calogero-Vasiliev algebra. For higher $\lambda$ values, they are shown to provide in their bosonic Fock space representation some interesting applications to supersymmetric quantum mechanics and some variants thereof: an algebraic realization of supersymmetric quantum mechanics for cyclic shape invariant potentials of period $\lambda$, a bosonization of parasupersymmetric quantum mechanics of order $p = \lambda-1$, and, for $\lambda=3$, a bosonization of pseudosupersymmetric quantum mechanics and orthosupersymmetric quantum mechanics of order two.