Ground algebras for superanalysis
type: Item
instance of: scholarly article
title: Ground algebras for superanalysis
zbMATH Open document ID: 0681.15014
DOI: 10.1016/0034-4877(91)90047-Q
published in: Reports on Mathematical Physics
publication date: 1992-09-27T00:00:00Z
review text: It is generally agreed now that any reasonable version of superanalysis should include an algebra serving as a supply of odd (anticommuting) constants (so to say, `superconstants'); the presence of such an algebra is inevitable when dealing with supersymmetry and many other matters of primary importance from the point of view of intended applications [see, e.g., \textit{A. Rogers}, Commun. Math. Phys. 105, 375-384 (1986; Zbl 0602.58003)]. Such an algebra should be a `Grassmannian' one, and there are numerous attempts to make this latter term precise. The most well- known among algebras of super-constants exploited so far are: the usual finite-dimensional Grassmann algebras; the Banach-Grassmann algebras [cf. \textit{A. Jadczyk} and \textit{K. Pilch}, ibid. 78, 373-390 (1981; Zbl 0464.58006)], the DeWitt-Fréchet algebra [cf. \textit{B. S. DeWitt}, Supermanifolds (1984; Zbl 0551.53002)], a nuclear LB-algebra introduced by \textit{Sh. Nagamachi} and \textit{Y. Kobayashi} [Lett. Math. Phys. 14, 15- 23 (1987; Zbl 0638.46054)] etc. The present paper is an attempt to systematize all those examples and propose more. The notion `supernumber algebra' is given a precise meaning: so is called a Hausdorff topological associative unital graded commutative algebra decomposing into the direct sum of the scalar field and the closed ideal topologically generated by the odd part. In particular, many new examples of Banach-Grassmann algebras are constructed, in addition to the only one previously known since the work of \textit{A. Rogers} [J. Math. Phys. 21, 1352-1365 (1980; Zbl 0447.58003)].
Mathematics Subject Classification ID: 15A75
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 46H20
Mathematics Subject Classification ID: 58C50
Mathematics Subject Classification ID: 81T60
zbMATH DE Number: 65384
zbMATH Keywords: superanalysis
zbMATH Keywords: superconstants
zbMATH Keywords: supersymmetry
zbMATH Keywords: Grassmann algebras
zbMATH Keywords: Banach- Grassmann algebras
zbMATH Keywords: DeWitt-Fréchet algebra
zbMATH Keywords: Supermanifolds
zbMATH Keywords: supernumber algebra
zbMATH Keywords: Hausdorff topological associative unital graded commutative
zbMATH Keywords: algebra
zbMATH Keywords: Hausdorff topological associative unital graded commutative algebra
author: Vladimir G. Pestov
reviewed by: Vladimir G. Pestov
MaRDI profile type: Publication
cites work: Q3344780
cites work: Q5531991
cites work: Q4136031
cites work: Topoi. The categorial analysis of logic
cites work: Q5573683
full work available at URL: 0034-4877(91)90047-q
OpenAlex ID: W2001589594
description: scientific article; zbMATH DE number 65384