Monoïdes de Clifford et déformations d'algèbres de Clifford. (Clifford monoids and deformations of Clifford algebras)
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.
scientific article; zbMATH DE number 4091678
scientific article