Covariant Differential and Integral Calculi for Lattice (l,q)-deformed Fields

Abstract

Using the Hecke $\hat R$-matrix, we give a definition of the lattice $(l,q)$-deformed $n$-component boson and Grassmann fields. Here $l$ is a deformation parameter for the commutation relations of "values" of these fields in two arbitrary lattice sites and $q$ is a deformation parameter for $n$-component $q$-boson or $q$-Grassmann variable. In framework of the Wess-Zumino approach to the noncommutative differential calculus the commutation relations between differentials and derivatives of these fields are determined. The $SL_q(n,C)$-invariant generalization of the Berezin integration for the lattice $n$-component $(l,q)$-Grassmann field is suggested. We show that the Gaussian functional integral for this field is expressed through the $(l,q)$-deformed counterpart of the Pfaffian.