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 Mat. Zametki, 1974, Volume 16, Issue 1, Pages 65–74 (Mi mz7436)

Automorphisms of the tensor product of Abelian and Grassmannian algebras

V. F. Pakhomov

M. V. Lomonosov Moscow State University

Abstract: We consider an algebra $\mathfrak B_{n,m}$, over the field $R$ with $n+m$ generators $x_1,…,x_n,\xi_1,…,\xi_m$, 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$'$}
\{\xi_i,\xi_j\}\equiv\xi_i\xi_j+\xi_j\xi_i=0, \tag{2$'$} \end{gather}
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.

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English version:
Mathematical Notes, 1974, 16:1, 624–629

Bibliographic databases:

UDC: 512

Citation: V. F. Pakhomov, “Automorphisms of the tensor product of Abelian and Grassmannian algebras”, Mat. Zametki, 16:1 (1974), 65–74; Math. Notes, 16:1 (1974), 624–629

Citation in format AMSBIB
\Bibitem{Pak74} \by V.~F.~Pakhomov \paper Automorphisms of the tensor product of Abelian and Grassmannian algebras \jour Mat. Zametki \yr 1974 \vol 16 \issue 1 \pages 65--74 \mathnet{http://mi.mathnet.ru/mz7436} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=357513} \zmath{https://zbmath.org/?q=an:0327.15032} \transl \jour Math. Notes \yr 1974 \vol 16 \issue 1 \pages 624--629 \crossref{https://doi.org/10.1007/BF01098815} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. D. A. Leites, “Ob odnom analoge opredelitelya”, UMN, 30:3(183) (1975), 156–156
2. B. L. Aneva, S. G. Mikhov, D. Ts. Stoyanov, “Some representations of the conformal superalgebra”, Theoret. and Math. Phys., 27:3 (1976), 502–508
3. V. P. Akulov, D. V. Volkov, V. A. Soroka, “Generally covariant theories of gauge fields on superspace”, Theoret. and Math. Phys., 31:1 (1977), 285–292
4. J. H. Bernstein, D. A. Leites, “Integral forms and the Stokes formula on supermanifolds”, Funct. Anal. Appl., 11:1 (1977), 45–47
5. D. A. Leites, “Introduction to the theory of supermanifolds”, Russian Math. Surveys, 35:1 (1980), 1–64
6. V. P. Akulov, I. A. Bandos, V. G. Zima, “Nonlinear realization of extended superconformal symmetry”, Theoret. and Math. Phys., 56:1 (1983), 635–642
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