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Journal of Lie Theory 14 (2004), No. 1, 073--109
Copyright Heldermann Verlag 2004
On Compactification Lattices of Subsemigroups of SL(2, R)
Brigitte E. Breckner
Babes-Bolyai University, Faculty of Mathematics and Computer Science,
Str. M. Kogalniceanu 1, 3400 Cluj-Napoca, Romania,
brigitte@math.ubbcluj.ro
Wolfgang A. F. Ruppert
Institut für Mathematik und Angewandte Statistik, Universität für Bodenkultur,
Peter Jordanstr. 82, 1190 Wien, Austria,
ruppert@edv1.boku.ac.at
Using the tools introduced in a previous article of the authors
[ J. Lie Theory 11 (2001), 559--604]
we investigate topological semigroup compactifications of closed connected
submonoids with dense interior of Sl(2, R). In particular, we show that the
growth of such a compactification is always contained in the minimal ideal,
and describe the subspace of all minimal idempotents (typically a two-cell)
and the maximal subgroups (these are always isomorphic to a compactification
of R). For a large class of such semigroups we give explicit constructions
yielding all possible topological semigroup compactifications and determine
the structure of the compactification lattice.
Keywords: Bohr compactification, lattice of compactifications, asymptotic
homomorphism, subsemigroups of Sl(2, R), Lie semigroups, Lie semialgebras,
diamond product, rectangular domain, umbrella set, divisible semigroup,
UDC semigroup.
MSC 2000: 22E15, 22E46, 22A15, 22A25.
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