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A generalization of super-Lie algebras is presented. It is then shown thatall known examples of fractional supersymmetry can be understood in thisformulation. However, the incorporation of three dimensional fractionalsupersymmetry in this framework needs some care. The proposed solutions leadnaturally to a formulation of a fractional supersymmetry starting from anyrepresentation D of any Lie algebra g. This involves taking the Fth-roots of Din an appropriate sense. A fractional supersymmetry in any space-time dimensionis then possible. This formalism finally leads to an infinite dimensionalextension of g, reducing to the centerless Virasoro algebra when g=sl(2,R).Comment: 23 pages, 1 figure, LaTex file with epsf.st

Fractional supersymmetry and Fth-roots of representations