Automorphisms of the tensor product of Abelian and Grassmannian algebras

Abstract

We consider an algebraB _n,m , over the field R with n+m generators x_i,..., x_n, ξ₁,..., η_m, satisfying the following relations:

$$\left[ {x_k ,x_l } \right] \equiv x_k x_l - x_l x_k = 0,[x_k ,\xi _i ] = 0,$$

((1'))

,

$$\left\{ {\xi _i ,\xi _j } \right\} \equiv \xi _i \xi _j + \xi _j \xi _i = 0$$

((2'))

, where k,l =1, ..., n and i, j=1,..., m. In this algebra we define differentiation, integration, and also a group of automorphisms. We obtain an integration equation invariant with respect to this group, which coincides in the case m=0 with the equation for the change of variables in an integral, an equation whichis well known in ordinary analysis; in the case n=0 our equation coincides with F. A. Berezin's result [1, 3] for integration over a Grassman algebra.