The Clifford Algebra and the Group of Similitudes

Abstract

LetC(M, Q)be the Clifford algebra of an even dimensional vector spaceMrelative to a quadratic formQ. WhenQis non-degenerate, it is well known that there exists an isomorphism of the orthogonal groupO(Q)onto the group of those automorphisms ofC(M, Q)which leave invariant the spaceM⊂C(M, Q). These automorphisms are inner and the group of invertible elements ofC(M, Q)which define such inner automorphisms is called the Clifford group.If instead of the groupO(Q)we take the group of similitudesγ(Q)or even the group of semi-similitudesΓγ(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 elementsC+(M, Q)⊂C(M, Q). Each one of the automorphisms ofC+(M, Q)so defined can be extended, as it is shown here (Theorem 2), to an inner automorphism ofC(M, Q), although the extension is not unique.