The structure of smooth mappings over Weil algebras and the category of manifolds over algebras
type: Item
instance of: scholarly article
title: The structure of smooth mappings over Weil algebras and the category of manifolds over algebras
zbMATH Open document ID: 0985.58001
published in: Lobachevskii Journal of Mathematics
publication date: 2000-07-25T00:00:00Z
full work available at URL: https://eudml.org/doc/225392
full work available at URL: shurygin.htm
review text: The general theory of manifolds over algebras was studied by A. P. Shirokov, J. Vanžura, V. V. Vishnevskiĭ, G. I. Kruchkovich, and others. \textit{A. P. Shirokov} [Tr. Geom. Semin. 1, 425-456 (1966; Zbl 0198.54601)] proved that structures of finite-dimensional manifolds over local algebras arise on tangent bundles and bundles of infinitely near points in the sense of A. Weil. Various types of manifolds over algebras were defined in the case of infinite-dimensional manifolds and algebras. Consider a finite-dimensional, local, associative, commutative and unital algebra \textbf{A} and finite-dimensional vector spaces, which are also \textbf{A}-modules. A map between such modules is called \textbf{A}-smooth if it is smooth and its differential is \textbf{A}-linear at every point of its domain. A smooth manifold modeled on an \textbf{A}-module is called an \textbf{A}-manifold if it has an atlas with \textbf{A}-smooth coordinate transformations. In this paper, the author studies the local expression of an \textbf{A}-smooth mapping from an open subset of the module \({\mathbf A}^n\) to the module \({\mathbf A}^k\). Next, in the case when a local algebra \textbf{A} is the semidirect sum \(\overline{\mathbf A} \oplus {\mathbf I}\) of a subalgebra \(\overline{\mathbf A}\) and an ideal \textbf{I}, the expression of an \(\overline{\mathbf A}\)-smooth mapping from an open subset \(\overline{\mathbf U}\) of \(\overline{\mathbf A}^n\) to \({\mathbf A}^k\) is established and it is proven that such a mapping can be prolonged to an \textbf{A}-smooth mapping between domains of \({\mathbf A}^n\) and \({\mathbf A}^k\). This result allows to construct a functor from the category of \(\overline{\mathbf A}\)-manifolds to the category of \textbf{A}-manifolds that is a generalization of the Weil functor of \textbf{A}-prolongation. Also, the local expression of an \textbf{A}-smooth mapping between \textbf{A}-modules of the form \({\mathbf A}^n \oplus \mathbb{R}^m\) is given. The author prese…
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 58A05
Mathematics Subject Classification ID: 53C12
zbMATH DE Number: 1463550
zbMATH Keywords: local Weil algebra
zbMATH Keywords: mapping over algebra
zbMATH Keywords: A-smooth map
zbMATH Keywords: manifold over algebra
zbMATH Keywords: A-manifold
author: Vadim V. jun. Shurygin
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1463550