Para-Grassmann Extensions of the Virasoro Algebra

Abstract

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.

Paragrassmann differential calculus

“…In fact, our aim was to generalize the Grassmann calculus and to apply the PGA to describing particles with paragrassmann variables, fractional spin and statistics, para-conformal and para-Virasoro algebras, etc. (the last topic is detailly treated in our next paper [21]). Up to now, the relations between PGA and quantum groups, being themselves interesting and beautiful, were not very helpful in these applications.…”

Toeplitz operators over the paragrassmann unit disk

“…In [4] it was introduced the para-superplane, which consists of z = (z, θ), where z ∈ C and θ is the generator of the real paragrassmann algebra and any function on the para-superplane has the form…”

Parageneralization of Peierls Bracket Quantization

“…In view of Eq. (19), P is a subalgebra (subring) of the algebra (ring) of polynomials T in θ αµ I . T is used as a larger space in which the calculations are performed.…”