1993/12/10 by A. T. Filippov, Filippov, A. T., A. B. Kurdikov +1
Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #hep-th
paper · pdf · doi:10.48550/arxiv.hep-th/9312081
12 pages
arxiv created 1993/12/10 · arxiv updated 2009/11/30
This is a brief review of our recent work attempted at a generalization of the Grassmann algebra to the paragrassmann ones. The main aim is constructing an algebraic basis for representing `fractional' symmetries appearing in 2D integrable models and also introduced earlier as a natural generalization of supersymmetries. We have shown that these algebras are naturally related to quantum groups with q = \rm root of unity. By now we have a general construction of the paragrassmann calculus with one variable and preliminary results on deriving a natural generalization of the Neveu--Schwarz--Ramond algebra. The main emphasis of this report is on a new general construction of paragrassmann algebras with any number of variables, N. It is shown that for the nilpotency indices (p + 1) = 3, 4, 6 the algebras are almost as simple as the Grassmann algebra (for which (p + 1) = 2). A general algorithm for deriving algebras with arbitrary p and N is also given. However, it is shown that this algorithm does not exhaust all possible algebras, and the simplest example of an `exceptional' algebra is presented for p = 4, N = 4.