Automorphisms of the tensor product of Abelian and Grassmannian algebras

V. F. Pakhomov

Matematičeskie zametki, Tome 16 (1974) no. 1, pp. 65-74

Résumé

We consider an algebra $\mathfrak B_{n,m}$, over the field $R$ with $n+m$ generators $x_1,\dots,x_n,\xi_1,\dots,\xi_m$, satisfying the following relations: \begin{gather} [x_k,x_l]\equiv x_kx_l-x_lx_k=0,\quad[x_k,\xi_i]=0, \tag{1</nomathmode><mathmode>$'$} \{\xi_i,\xi_j\}\equiv\xi_i\xi_j+\xi_j\xi_i=0, \tag{2$'$} \end{gather}</mathmode><nomathmode> where $k,l=1,\dots,n$ and $i,j=1,\dots,m$. 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.