Paragrassmann Integral, Discrete Systems and Quantum Groups

This report is based on review paper [1]. Some aspects of differential and integral calculi on generalized grassmann (paragrassmann) algebras are considered. The integration over paragrassmann variables is applied to evaluate the partition function for the Z p+1 Potts model on a chain. Finite dimensional paragrassmann representations for GL q (2) are constructed. Generalizations of grassmann algebras have been considered by many authors (see e.g. [2]-[6] and references therein) ¿from different points of view. Those generalizations were stimulated by investigations in 2D conformal field theories [5, 8], anionic models and topological field theories which led to consideration of unusual statistics. The last one includes not only well known parastatistics [4] but also fractional and braid statistics [6, 7]. We note also attempts to generalize supersymmetry to parasupersymmetry [3, 5]. Our construction of paragrassmann algebras (PGA) with many variables [9, 10] is a direct generalization of the Weyl construction [11] for the grassmann Heisenberg-Weyl algebra. Let us consider the algebra Π p+1 with two nilpotent generators θ and ∂: θ p+1 = 0 = ∂ p+1 , θ p = 0 , ∂ p = 0 , (1) where p is a positive integer (usually called the order of parastatistics). The defining relation for this algebra is chosen such that one can push ∂ to the right: