INTRODUCTION

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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