A Criterion for the Minimal Closedness of the Lie Subalgebra Corresponding to a Connected Nonclosed Lie Subgroup
A Lie subalgebra of a Lie algebra is said to be minimally closed (after A. Malcev (1942)) if the corresponding connected Lie subgroup is closed in the simply connected Lie group determined by . The aim of this paper is to prove the following theorem:
Let be any connected (not necessarily closed) Lie subgroup of a Lie group . Denote by , , and the Lie algebras of , of its closure , and of , respectively. If there exists a Lie subalgebra such that (a) , (b) , then is minimally closed.
As a corollary we obtain that if is finite, then no such a Lie subalgebra exists provided that is nonclosed.
The proof is carried out on the ground of the theory of Lie algebroids and by using some ideas from the theory of transversally complete foliations.
1991 Mathematics Subject Classification: 17B05, 22E15, 22E60.