Lie subgroups and Lie correspondence

For a Lie subalgebra h of the Lie algebra of G, left translations of h form an involutive distribution. The Frobenius theorem integrates the distribution through the identity, producing a unique connected immersed Lie subgroup with that tangent algebra. The subgroup generated by exp h describes the same object. Its intrinsic manifold topology is part of the construction; it is not automatically the topology inherited as a subset of G.

An irrational line in the Lie algebra of a two-dimensional torus illustrates the issue. Exponentiating it gives an injectively immersed real line with dense image in the torus. It is not an embedded copy of the real line with the torus subset topology and is not a closed subgroup. By contrast, a rational-slope line closes into a circle. The tangent dimension is one in both cases, so tangent data alone cannot decide whether the image is closed.

Every finite-dimensional real Lie algebra is the Lie algebra of a connected simply connected real Lie group, unique up to an isomorphism inducing the specified algebra identification. For connected simply connected G, every Lie algebra homomorphism from its algebra to the algebra of another Lie group H integrates uniquely to a smooth group homomorphism G→H. The target does not need to be simply connected. The source assumption removes ambiguity caused by continuing a local map around nontrivial loops.

For a connected source that is not simply connected, first integrate on its universal covering group. The map descends to the original source exactly when it kills the covering kernel. That kernel is discrete and central, and its image is not determined merely by checking the infinitesimal bracket equation. For example, all scalar linear maps between one-dimensional abelian Lie algebras preserve brackets, but only particular scalars respect the circle's integer periods.

The additive real line and the circle both have a one-dimensional abelian Lie algebra, but their topology differs. More generally, a connected Lie group is a quotient of its simply connected covering group by a discrete central subgroup. This preserves the Lie algebra while changing global periods, fundamental groups, and which representations descend. It does not say every abstract central subgroup is an admissible quotient: discreteness supplies the appropriate covering behavior in this description.

When comparing two integrations, list the simply connected group, the chosen discrete kernel, and the proposed map's action on that kernel. For a subgroup inside an existing ambient group, separately ask whether its immersed image is closed. These questions are related through topology but are not interchangeable. A dense immersed subgroup can be the correct analytic subgroup for a subalgebra while failing to provide a Hausdorff homogeneous-space quotient with the ordinary subset quotient topology.