Semigroups in Lie groups, semialgebras in Lie algebras

Abstract

Consider a subsemigroup of a Lie group containing the identity and being ruled by one-parameter semigroups near the identity.We associate with it the set W of its tangent vectors at the identity and obtain a subset of the Lie algebra L of the group.The set W has the following properties: (i) W + W = W, (ii) R + -W c W, (iii) W'= W, and, the crucial property, (iv) for all sufficiently small elements x and y in W one has x * y = x + y + \[x, y] + (Campbell-Hausdorff!) e W. We call a subset W of a finite-dimensional real Lie algebra L a Lie semialgebra if it satisfies these conditions, and develop a theory of Lie semialgebras.In particular, we show that a subset W satisfying (i)-(iii) is a Lie semialgebra if and only if, for each point x of W and the (appropriately defined) tangent space Tx to W in x, one has [x, Tx] c Tx. (The Lie semialgebra W of a subgroup is always a vector space, and for vector spaces W we have Tx = W for all x in W, and thus the condition reduces to the old property that W is a Lie algebra.)In the introduction we fully discuss all Lie semialgebras of dimension not exceeding three.Our methods include a full duality theory for closed convex wedges, basic Lie group theory, and certain aspects of ordinary differential equations.