Hilbert space methods in the theory of Lie algebras

Abstract

Introduction. In the study of the complex finite-dimensional semi-simple Lie algebras a crucial role is played by the fundamental bilinear form (x, y) = Tr(ad(x)ad(y)). Since the definition is meaningless when the restriction of finite-dimensionality is removed, if any of the highly desirable properties of the form are to be retained in this case they must necessarily be given a priori. By reconsidering the finite-dimensional situation it is possible to formulate suitable conditions in a more convenient form. To see this let P be a complex finite-dimensional semi-simple Lie algebra and let Po be a compact real form for P with cr as the associated involution (conjugation). If we let x* = —a(x) and (x, y) = (x, y*) then 7 becomes a finite-dimensional Hilbert space, the mapping x into x* is a Hilbert space conjugation, and the connecting property ([x, y], z) = (y, [x*, z]) holds for all x, y, z. An P* algebra as defined here is simply a Lie algebra whose vector space is a Hilbert space such that the connecting property above holds. This paper is a study of such algebras with emphasis, of course, on the infinite-dimensional ones. For finite dimensions nothing new is obtained and it is shown here that in this case every semisimple P* algebra arises essentially from a construction like that above (see the remark after 2.5). There is an associative algebra analogue of this problem in the paper of Ambrose [l ] on TZ* algebras and some of his results are used here. Any H* algebra gives rise to an P* algebra by letting [x, y]=xy — yx and the only known examples of P* algebras are those obtained as Lie subalgebras of H* algebras. The main result of this paper is a classification of the (separable) simple P* algebras which have Cartan decompositions (see §2) and it is shown that this class coincides with the simple self-adjoint Lie subalgebras of a (separable) simple H* algebra. The results turn out to be the natural extensions of the finite-dimensional theory. Associated with each of the Lie algebras considered here there is a gener-