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The Clifford Algebra and the Group of Similitudes

Published online by Cambridge University Press:  20 November 2018

Maria J. Wonenburger
Maria J. Wonenburger*
Affiliation:
Queen's University, Kingston C.S.I.C., Madrid
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Let C(M, Q) be the Clifford algebra of an even dimensional vector space M relative to a quadratic form Q. When Q is non-degenerate, it is well known that there exists an isomorphism of the orthogonal group O(Q) onto the group of those automorphisms of C(M, Q) which leave invariant the space M ⊂ C(M, Q). These automorphisms are inner and the group of invertible elements of C(M, Q) which define such inner automorphisms is called the Clifford group.

If instead of the group O(Q) we take the group of similitudes γ(Q) or even the group of semi-similitudes Γγ(Q), it is possible to associate in a natural way with any element of these groups an automorphism or semi-automorphism, respectively, of the subalgebra of even elements C+(M, Q) ⊂ C(M, Q). Each one of the automorphisms of C+(M, Q) so defined can be extended, as it is shown here (Theorem 2), to an inner automorphism of C(M, Q), although the extension is not unique.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 45 - 59
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Bourbaki, N., Algèbre (Paris: Hermann et Cie, 1947), chapter m.Google Scholar
2. Chevalley, C., Théorie des groupes de Lie, II (Paris: Hermann et Cie, 1951).Google Scholar
3. Chevalley, C., The algebraic theory of spinors (New York: Columbia University Press, 1954).Google Scholar
4. Chevalley, C., The construction and study of certain important algebras, Math. Soc. Japan (1955).Google Scholar
5. Dieudonné, J., Sur les groupes classiques (Paris: Hermann et Cie, 1948).Google Scholar
6. Dieudonné, J., Sur les multiplicateurs des similitudes, Rend. Cire. Mat. Palermo, 3 (1954), 398–408.Google Scholar
7. Dieudonné, J., La géométrie des groupes classiques (Berlin: Springer-Verlag, 1955).Google Scholar
8. Dieudonné, J., On the automorphisms of the classical groups, Mem. Amer. Math. Soc, No. 2 (1951).Google Scholar
9. Eichler, M., Quadratische Formen und Orthogonale Gruppen (Berlin: J. Springer, 1952).Google Scholar