Paragrassmann algebras with one and many paragrassmann variables are considered from the algebraic point of view without using the Green ansatz. A differential operator with respect to paragrassmann variable and a covariant para-super-derivative are introduced giving a natural generalization of the Grassmann calculus to a paragrassmann one. Deep relations between paragrassmann algebras and quantum groups with deformation parameters being roots of unity are established.

This paper significantly extends and generalizes our previous paper [1]. Here we discuss explicit general constructions for paragrassmann calculus with one and many variables. For one variable, nondegenerate differentiation algebras are identified and shown to be equivalent to the algebra of (p+1)×(p+1) complex matrices. If (p+1) is prime integer, the algebra is nondegenerate and so unique. We then give a general construction of the many-variable differentiation algebras. Some particular examples are related to the multi-parametric quantum deformations of the harmonic oscillators. * Address until Dec.22, 1992: Yukawa Inst. Theor. Phys., Kyoto Univ.,

The paragrassmann differential calculus is briefly reviewed. Algebras of the transformations of the para-superplane preserving the form of the parasuperderivative are constructed and their geometric meaning is discussed. A new feature of these algebras is that they contain generators of the automorphisms of the paragrassmann algebra (in addition to Ramond-Neveu-Schwarz -like conformal generators). As a first step in analyzing these algebras we introduce more tractable multilinear algebras not including the new generators. In these algebras there exists a set of multilinear identities based on the cyclic polycommutators. Different possibilities of the closure are therefore admissible. The central extensions of the algebras are given. Their number varies from 1 to [ p+1 2 ] depending on the form of the closure chosen. Finally, simple explicit examples of the paraconformal transformations are given.