Fractional supersymmetry and \(F\)th-roots of representations (Q2738062)

The paper deals with generalized super-Lie algebras. For any positive integer \(F\) a fractional super-Lie algebra (\(F\)-Lie algebra) has been constructed. If \(F=1\), then an \(F\)-Lie algebra is a Lie algebra. If \(F=2\), then an \(F\)-Lie algebra is a super-Lie algebra. These algebras allow to generalize supersymmetry in the sense of \(F\)-th roots of representations. The formalism was built using an arbitrary representation \(\varphi\) of any semisimple complex Lie algebra \({\mathfrak g}\). It also allows to obtain the special infinite dimensional Lie algebra \(V({\mathfrak g})\) containing \({\mathfrak g}\) as a subalgebra. If \({\mathfrak g}={\mathfrak g}{\mathfrak l}(2,\mathbb{R})\), then \(V({\mathfrak g})\) is the centerless Virasoro algebra.