Mid-reversibility properties of semigroup actions on homogeneous spaces

• Published in: Semigroup forum Vol. 104; no. 3; pp. 689 - 703
• Main Authors: Reis, Ronan A., San Martin, Luiz A. B., Rocha, Victor H. L.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-06-2022; Springer Nature B.V
• Subjects: Algebra; Lie groups; Mathematics; Mathematics and Statistics; Research Article; Semigroups; Subgroups; Flag manifolds; Semigroups; Semi-simple Lie groups; Mid-reversibility
• ISSN: 0037-1912; 1432-2137
• DOI: 10.1007/s00233-022-10275-5

In this paper we study mid-reversibility of subsemigroups acting on homogeneous spaces. The mid-reversor set of a subsemigroup is defined and it is described in terms of the invariant control sets for semigroups acting on certain homogeneous spaces. Let
G
be a connected noncompact semi-simple Lie group and
L
a subgroup of
G
. Assume that
S
is a subsemigroup of
G
with nonempty interior. We characterize the mid-reversibility of the
S
-action on
G
/
L
in terms of the actions of
S
and
L
on the flag manifolds of
G
. We show that the mid-reversibility of
S
in
G
/
L
is related to the reversibility of
S
in
G
/
L
. We also present sufficient conditions for
S
to generate
G
if
S
is mid-reversible in
G
/
L
.