Algebraic realization of supersymmetric quantum mechanics for cyclic shape invariant potentials

We study in detail the spectrum of the bosonic oscillator Hamiltonian associated with the C-3-extended oscillator algebra A(alpha 0 alpha 1)((3)) where C-3 denotes a cyclic group of order three, and classify the various types of spectra in terms of the algebra parameters alpha(0), alpha(1). In such a classification, we identify those spectra having: an infinite number of periodically; spaced levels, similar to those of cyclic shape invariant potentials of period three. We prove that the hierarchy of supersymmetric Hamiltonians and supercharges, corresponding to the latter, can be realized in terms of some appropriately chosen A(alpha 0 alpha 1)((3)) algebras, and of Pauli spin matrices. Extension to period-lambda spectra in terms of Cx-extended oscillator algebras is outlined.