Paragrassmann Algebras with Many Variables

• Published in: arXiv.org
• Main Authors: Filippov, A T, Kurdikov, A B
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 10.12.1993
• Subjects: Algebra; Algorithms; Two dimensional models; Vector spaces
• ISSN: 2331-8422

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 \(p = 4, N = 4\).