Covariant differential calculus on the quantum exterior vector space

Abstract

We formulate a differential calculus on the quantum exterior vector space spanned by the generators of a non-anticommutative algebra satisfying

$$\begin{gathered} r^{ij} \equiv \theta ^i \theta ^j + B_{kl}^{ij} \theta ^k \theta ^l = 0i,j = 1,2,...,n. \hfill \\ and \hfill \\ (\theta ^i )^2 = (\theta ^j )^2 = ... = (\theta ^n )^2 = 0, \hfill \\ \end{gathered} $$

whereB ^ij_kl is the most general matrix defined in terms of complex deformation parameters. Following considerations analogous to those of Wess and Zumino, we are able to exhibit covariance of our calculus under (n/2)+1 parameter deformation ofGL(n) and explicitly check that the non-anticommutative differential calculus satisfies the general constraints given by them, such as the “linear” conditiondr ^ij≃0 and the “quadratic” conditionr ^ij x ⁿ≃0 wherex ⁿ=dλ _n are the differentials of the variables.