This work is devoted to the study of closed invariant subsemigroups of Lie
groups which are topologicaUy generated by one-parameter semigroups. We call
these semigroups invariant Lie semigroups. Dropping the invariance condition
we speak of Lie semigroups. These semigroups are important for various reasons.
An invariant order on a Lie group G is a partial order on G which is
invariant under both left and right shifts. Then the set
S := {g £ G : 1 g)
is an invariant submonoid of G with H(S) = { l } . If, conversely, S C G is an
invariant submonoid with H(S) = {1} the prescription
9 g' if g'e gS
defines an invariant order on G. We say that is a continuous order if the
semigroup S is closed and topologicaUy generated by every neighborhood of 1.
According to a result of the author ([Ne91d]) a closed submonoid S of G with
H(S) = {1} defines a continuous invariant order if and only if S is an invariant
Lie semigroup. This is the connection of invariant Lie semigroups with these
orders. Invariant orders on Lie groups are studied in [Vin80], [0182a], [Pa81]
(in semi-simple groups), [Gi89] (in solvable groups) and by the author ([Ne91d],
[Ne91e], [Ne88]). One of the most interesting questions in this context is the
existence problem. When does a connected Lie group G admit a continuous
invariant order?
If G is a connected Lie group and A C G x G is the diagonal, then
M := (G x G)/A is a simple example of a symmetric space. If, in addition,
S C G is a Lie semigroup, then Si := (S x {1})A is a subsemigroup of G x G
which defines a G -invariant partial order on M by
(gug2)A (gi,g'2)A if (g'l9g2) e (gi,g2)S1.
So invariant Lie semigroups are closely related to ordered symmetric spaces. If,
more generally, S C G is a Lie semigroup, and H(S) := S H S"1 its group of
units, then the prescription
gH(S) g'H(S) if g' gS
Received by the editors November 5, 1990.
Research supported by a postdoctoral grant of the Deutsche Forschungsgemeinschaft.
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