I.—On the Geometry of Metrisable Lie Algebras

Synopsis

Metrisable Lie algebras have been defined by Tsou and Walker (1957). Their definition is adopted below in § I.

The object of the present paper is to display something of the geometrical background of such algebras, particularly for those taken over the field of real numbers.

§ I is introductory. In § 2 appears a statement, made as brief as possible because it is wholly classical, of the relationship between the vector-space of the Lie algebra L and the associated affine and projective spaces A and P. Some properties of metrisable Lie algebras are then examined in terms of the geometry of P, which provides an (n –I)-dimensional map of the n-dimensional algebra L. It is assumed throughout the paper that the Lie algebras under discussion are non-abelian, since the projective map of an abelian algebra presents nothing of interest.

As the present work is intended as no more than a preliminary, it is confined, so far as its applications are concerned, to a discussion of metrisable algebras of dimensions 3 and 4 and to one example of an algebra of dimension 6.

I have been privileged in preparing the paper to have access to the typescript of the paper by Tsou and Walker referred to above, and also to the doctoral thesis (1955) of the former. I am greatly indebted to them both, and also to Dr Paul Cohn and to a referee for suggestions regarding certain details of presentation.