LIE-GROUPS AND SUBSEMIGROUPS WITH SURJECTIVE EXPONENTIAL FUNCTION

A closed subsemigroup S in a Lie group G determines a closed convex we dge L(S) in the Lie algebra L(G) of G. If S is a subgroup, then L(S) i s the Lie algebra of S; in the general case L(S) is called the Lie wed ge of S. The subsemigroup S of a Lie group is called reduced, if it is closed and does not contain a nonsingleton normal subgroup. The expon ential function exp(G): L(G)-->G induces an exponential function exp(S ): L(S)-->S. The subsemigroup S is called exponential, respectively, w eakly exponential, if exp(S) L(S) = S, respectively, exp(S) L(S) = S. This definition applies, in particular, with S = G and allows the form ulation of the following problems in the structure theory of Lie group s:Problem 1. Characterize all exponential Lie groups. Problem 2. Chara cterize all weakly exponential Lie groups. Problem 1 is open while Pro blem 2 is solved. We address the analogous problems for reduced subsem igroups of Lie groups: Problem 3. Characterize all exponential reduced subsemigroups of Lie groups. Problem 4. Characterize all weakly expon ential reduced subsemigroups of Lie groups. Problems 3 and 4 are compl etely settled in this memoir. In the process it is shown that all weak ly exponential reduced subsemigroups are exponential and that the Lie wedge L(S) of an exponential reduced subsemigroup S of a Lie group G i s a Lie semialgebra in G(G). Lie semialgebras have been classified.