The Duality Between Subsemigroups of Lie Groups and Monotone Functions

In this paper we give a characterization of those convex cones $W$ in the Lie algebra ${\mathbf {L}}(G)$ of a connected Lie group $G$ which are global in $G$, i.e. for which there exists a closed subsemigroup $S$ in $G$ having $W$ as its tangent wedge ${\mathbf {L}}(S)$. The main result is the Characterization Theorem II.12. We also prove in Corollary II.6 that each germ of a strictly $W$-positive function belongs to a global function if there exists at least one strictly $W$-positive function. We apply the Characterization Theorem to obtain some general conditions for globality and to give a complete description of the global cones in compact Lie algebras.