Invariant convex cones and causality in semisimple Lie algebras and groups

Abstract

The convex cones in a simple Lie algebra G invariant under the adjoint group G of G are studied. Using a earlier abstract classification of such cones, we find explicit algebraic presentations of such cones in all the classical hermitian symmetric Lie algebras. (Nontrivial such cones exist only in these cases.) The G-orbits in such cones are listed. The notion of a temporal action of a Lie group with an invariant causal orientation upon a causally oriented manifold is defined. The canonical actions of such classical groups G as above on the S̆hilov boundaries of the associated (tube-type) hermitian symmetric spaces are shown to be temporal actions. Corollaries are (1) the existence of nontrivial (Lie) semigroups S in the infinite-sheeted coverings G ̃ of G, which are invariant under conjugation by G ̃ and satisfy S ∩ S −1 = { e}, and (2) the global causality (i.e. no “closed time-like curves”) of such covering groups G ̃ .