Abstract:
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.
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