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The Hamiltonian for a fractional supersymmetric oscillator is derived fromthree approaches. The first one is based on a decomposition in which a Q-uongives rise to an ordinary boson and a k-fermion (a k-fermion being an objectinterpolating between boson and fermion). The second one starts from ageneralized Weyl-Heisenberg algebra. Finally, the third one relies on thequantum algebra Uq(sl(2)) where q is a root of unity.Comment: 14 pages, LaTex file. Paper written from a lecture presented (by  M.R.K.) at the Sixth International School on Theoretical Physics ``Symmetry  and Structural Properties of Condensed Matter'', a school devoted to the  memory of Louis Michel, Myczkowce (Poland), 31 August - 6 September 2000. To  be published in ``Symmetry and Structural Properties of Condensed Matter'',  eds. T. Lulek, B. Lulek and A. Wal (World Scientific, Singapore, 2001

ON FRACTIONAL SUPERSYMMETRIC QUANTUM MECHANICS: THE FRACTIONAL SUPERSYMMETRIC OSCILLATOR