RMC - Vol. 4, Num. 2, 3

Full paper in PDF:
$% J. Kubarski, A criterion for the minimal closedness of the Lie subalgebra corresponding to a connected nonclosed Lie subgroup, Rev. Mat. Univ. Complut. Madrid 4 (1991), no. 2, 3, 159–176.%$

A Criterion for the Minimal Closedness of the Lie Subalgebra Corresponding to a Connected Nonclosed Lie Subgroup

Jan KUBARSKI

Institute of Mathematics
Technical University of Lodz — Poland

Received: March 28, 1990

ABSTRACT

A Lie subalgebra $h$ of a Lie algebra $g$ 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 $g$ . The aim of this paper is to prove the following theorem:

Let $H < G$ be any connected (not necessarily closed) Lie subgroup of a Lie group $G$ . Denote by $h$ , $h$ , and $g$ the Lie algebras of $H$ , of its closure $H$ , and of $G$ , respectively. If there exists a Lie subalgebra $c< g$ such that (a) $c+ h= g$ , (b) $c /~\ h= h$ , then $h$ is minimally closed.