On the group of similitudes and its projective group

Abstract

[Abstract]: Given a vector space nr over a field, with a non-degenerate quadratic form Q, we consider the linear transformations S which take any vector x ϵ nr into a vector xS such that Q(xS).= ρQ(x), where ρ ≠ 0 only depends on S. Such transformations are called similitudes. The group of similitudes seems to be more natural than the orthogonal group. The projective group of similitudes can be realized as a group of automorphisms of the subalgebra C+ of the Clifford algebra. These automorphisms can be characterized in a similar way as the automorphisms corresponding to the orthogonal group have been characterized before. The study of the elements of the group of similitudes as automorphisms of the whole Clifford algebra allow us to find the structure of this group, when the index of Q is greater than 0.

Under this assumption the automorphisms of the group of similitudes and its projective group are found using the known results about the automorphisms of the orthogonal or rotation groups and their projective groups. The methods which lead to the determination of these automorphisms are a combination of Dieudonné's methods for finding out the automorphisms of the orthogonal group using the structure of these groups and Rickart's methods which consist in distinguishing the extremal involutions from non-extremal involutions by group theoretic properties.

In order to study the properties of certain similitudes a correspondence has been established between the centralizer of such similitudes in the orthogonal group and the unitary group of a hermitian form over a field. This suggests the definition of a certain algebra attached to a hermitian form over a field such that the relations between this algebra and the group of unitarian similitude are analogous to the relations of the Clifford algebra with the group of similitudes. We only mention here that in a similar way given a hermitian form over a division ring of quaternions one can connect with it a quadratic form (see [10]) and define an algebra corresponding to this hermitian form.

It was found convenient to establish first of all certain general properties of the Clifford algebra without any assumption on the non-degeneracy of Q which might have some interest in themselves. The equivalent known results for the Grassman algebra follow from ours when the quadratic form Q is identically 0.