On Sophus Lie's fundamental theorem

Abstract

Sophus Lie's third theorem asserts that for every finite dimensional Lie algebra there is a local Lie group with the given algebra as tangent algebra at the identity. In particular, each subalgebra of the Lie algebra of a local Lie group determines a local subgroup whose Lie algebra is the given one. Recent interest in subsemigroups and local subsemigroups of Lie groups motivate the investigation to which extent local semigroups in Lie groups are determined by infinitesimal generators. It is known that the set L( S) of tangent vectors at the identity of a local semigroup S with identity in a (local) Lie group G is an additively closed convex subset of the Lie algebra L( G) which is topologically closed and contains 0; its largest vector subspace H is a subalgebra and all automorphisms of L( G) of the form e ad x with x ϵ H leave L( S) invariant as a whole (with (ad x)( y) = [ x, y]). We call an additively and topologically closed convex subset W containing 0 in a Lie algebra L a Lie wedge if it satisfies e ad x W = W for all x ϵ W ∩ t- W. In this paper we show that Lie wedges are precisely the infinitesimal generators of local semigroups in Lie groups. More precisely, we show that in a Banach Lie algebra L for every Lie wedge W whose edge W ∩ − W has a closed vector space complement in L, there is an open ball B around 0 in L such that the Cambell Hausdorff series x ∗ y = x + y + 1 2 [x, y] + ··· converges absolutely for x, y ϵ B and there is a subset S ⊂ B with (i) 0 ϵ S, (ii) (S ∗ S) ∩ B ⊂S, and L(S) = W .