Let S be a locally compact
semigroup consisting of a dense connected Lie group G and its boundary L; and let e
be an idempotent in L. This paper is concerned with the proof of three
principal results: (1) If L = Ge is simply connected, then S is homeomorphic
to G∕Gr(e) × Gr(e)∕Gl(e) × Gl(e)−, where Gl(e)− is a connected locally
compact group with zero. (2) For any connected Lie group G and closed
normal subgroup H such that G∕H is simply connected and H is the direct
product of the multiplicative group of positive real numbers and a connected
compact group, there is a locally compact semigroup S which contains a
dense subgroup isomorphic to G whose boundary is a group isomorphic to
G∕H. (3) If G = V∗Gl(e), for some subspace V ⊂ G, then Ge is locally
compact if and only if there is a local cross-section to the global orbits of G at
e.
