Fractional supersymmetric quantum mechanics, topological invariants and generalized deformed oscillator algebras

Fractional supersymmetric quantum mechanics of order $\lambda$ is realized in terms of the generators of a generalized deformed oscillator algebra and a Z$_{\lambda}$-grading structure is imposed on the Fock space of the latter. This realization is shown to be fully reducible with the irreducible components providing $\lambda$ sets of minimally bosonized operators corresponding to both unbroken and broken cases. It also furnishes some examples of Z$_{\lambda}$-graded uniform topological symmetry of type (1, 1, ..., 1) with topological invariants generalizing the Witten index.