  A generalization of super-Lie algebras is presented. It is then shown that
all known examples of fractional supersymmetry can be understood in this
formulation. However, the incorporation of three dimensional fractional
supersymmetry in this framework needs some care. The proposed solutions lead
naturally to a formulation of a fractional supersymmetry starting from any
representation D of any Lie algebra g. This involves taking the Fth-roots of D
in an appropriate sense. A fractional supersymmetry in any space-time dimension
is then possible. This formalism finally leads to an infinite dimensional
extension of g, reducing to the centerless Virasoro algebra when g=sl(2,R).
