Monoïdes de Clifford et déformations d'algèbres de Clifford. (Clifford monoids and deformations of Clifford algebras)
type: Item
instance of: scholarly article
title: Monoïdes de Clifford et déformations d'algèbres de Clifford. (Clifford monoids and deformations of Clifford algebras)
zbMATH Open document ID: 0667.15024
DOI: 10.1016/0021-8693(87)90241-9
author: Jacques Helmstetter
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: The author like to point out that considerable modification is necessary to adopt the classical theory of Clifford groups in the case of C-algebra in a covering space. Accordingly, the important object is not the group of Clifford but a `monoïd', called as the monoïd of Clifford. The relation between the two objects in the classical case of a non- degenerate quadratic form on a vector space of finite dimension may be stated, in short, thus: on the one hand the Clifford group is the group of invertible elements of the `monoïds' of Clifford, on the other hand the `monoïd' of Clifford is the support of the Clifford group in the Clifford algebra. To start with, the author presents the filtrations of the algebra of Clifford, which forms the basis of the definition of a Clifford `monoïd'. Certain properties of the monoïds of Clifford, which are of importance in the context, are then studied. Let Q and \(Q'\) be two quadratic forms on the same vector space and let C(Q) and \(C(Q')\) be associated Clifford algebras respectively. Further let B be a linear bijective mapping of C(Q) in \(C(Q')\). In the vector space C(Q,B), a new inner product is defined. This is an algebra isomorphic to \(C(Q')\). Consequently one can define a Clifford group \(\Gamma\) (Q,B) isomorphic to \(\Gamma (Q')\) and the monoïd of Clifford X(Q,B) isomorphic to \(X(Q')\). The assertion is \(X(Q,B)=X(Q').\) Finally, the relation of this development with existing theories is discussed.
Mathematics Subject Classification ID: 15A66
zbMATH DE Number: 4091678
zbMATH Keywords: isomorphism
zbMATH Keywords: module
zbMATH Keywords: monoïd of Clifford
zbMATH Keywords: quadratic form
zbMATH Keywords: Clifford group
zbMATH Keywords: Clifford algebra
zbMATH Keywords: filtrations
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90241-9
OpenAlex ID: W2046877222
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description: scientific article; zbMATH DE number 4091678