A deformed q-calculus is developed on the basis of an algebraic structure involving graded brackets. A number operator and left and right shift operators are constructed for this algebra, and the whole structure is related to the *

We show that one-dimensional superspace is isomorphic to a non-trivial but consistent limit as $q\to-1$ of the braided line. Supersymmetry is identified as translational invariance along this line. The supertranslation generator and covariant derivative are obtained in the limit in question as the left and right derivatives of the calculus on the braided line.Comment: LateX file. 10 pages. To appear in Phys. Lett.

It is shown that fractional Z3-superspace is isomorphic to the q → exp(2πi/3) limit of the braided line. Z3-supersymmetry is identified as translational invariance along this line. The fractional translation generator and its associated covariant derivative emerge as the q → exp(2πi/3) limits of the left and right derivatives from the calculus on the braided line. Brackets and q-gradingOur aim here is to reformulate some results of a previous paper [1], where the structure of fractional supersymmetry was investigated from a group theoretical point of view, from a braided Hopf algebra approach. We shall not be concerned here with the possible applications of fractional supersymmetry and will refer instead to [1,2] for references on this aspect.We begin by defining the bracketwhere q and r are just arbitrary complex numbers. If we assign an integer grading g(X) to each element X of some algebra, such that g(1)=0 and g(XY ) = g(X) + g(Y ), for any X and Y , we can define a bilinear graded z-bracket as follows,.( 1.2) Here A and B are elements of the algebra, and of pure grade. The definition may be extended to mixed grade terms using the bilinearity. We also havesupplemented by [0] q ! = 1. When q is n-root of unity the previous grading scheme becomes degenerate, so that in effect the grading of an element is only defined modulo n. In this case also have [r] q = 0 when r modulo n is zero (r = 0).

A many variable q-calculus is introduced using the formalism of braided covector algebras. Its properties when certain of its deformation parameters are roots of unity are discussed in detail, and related to fractional supersymmetry. The special cases of two dimensional supersymmetry and fractional supersymmetry are developed in detail.

The origin of and distinction between the two deformation techniques currently in use for single particle paraoscillators is made clear. For paraoscillators deformed using the more familiar method, algebra deformation, (i) a single deformation function unifying parabosons and parafermions is provided and (ii) Fock space representations of all single paraoscillator systems as bilinear combinations of deformed oscillators are obtained. For the (distinct) Fock space deformation method, the deformed paraoscillator algebras are worked out in detail and their Casimir operators are given. Difficulties concerning the relationship between the single particle Calogero–Vasiliev oscillator and the single particle parabose system are resolved.