Homomorphisms and Coverings

Abstract

The results about Lie subgroups proved so far allow to obtain information about homomorphisms between Lie groups. The idea is that the graph of a homomorphism ϕ : G → H is a subgroup of the product group G × H isomorphic to G by the projection π ₁ : G × H → G, $\pi _{1} \left ( x,y\right ) =x$ and, conversely, if the graph of a map ϕ is a subgroup, then ϕ is a homomorphism. In case the homomorphism ϕ is continuous or differentiable, its graph has topological or differentiable properties. One of the applications obtained is the proof that any homomorphism between the Lie algebras “extends” to the groups when the domain is simply connected. Bringing together these extensions and the construction of a Lie group structure on the universal covering of a given group, one obtains a description of connected Lie groups from simply connected Lie groups. The isomorphism classes of connected and simply connected groups are in bijection with the isomorphism classes of finite dimensional Lie algebras.