information:classification, structural:scholar_edition
Homomorphisms and isomorphisms

If G and H are Lie groups, then a Lie group homomorphism f : G → H is a smooth group homomorphism. In the case of complex Lie groups, such a homomorphism is required to be a holomorphic map. However, these requirements are a bit stringent; every continuous homomorphism between real Lie groups turns out to be (real) analytic.

The composition of two Lie homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a category. Moreover, every Lie group homomorphism induces a homomorphism between the corresponding Lie algebras. Let ϕ: G → H {\displaystyle \phi \colon G\to H} be a Lie group homomorphism and let ϕ ∗ {\displaystyle \phi _{*}} be its derivative at the identity. If we identify the Lie algebras of G and H with their tangent spaces at the identity elements, then ϕ ∗ {\displaystyle \phi _{*}} is a map between the corresponding Lie algebras:

which turns out to be a Lie algebra homomorphism (meaning that it is a linear map which preserves the Lie bracket). In the language of category theory, we then have a covariant functor from the category of Lie groups to the category of Lie algebras which sends a Lie group to its Lie algebra and a Lie group homomorphism to its derivative at the identity.

Two Lie groups are called isomorphic if there exists a bijective homomorphism between them whose inverse is also a Lie group homomorphism. Equivalently, it is a diffeomorphism which is also a group homomorphism. Observe that, by the above, a continuous homomorphism from a Lie group G {\displaystyle G} to a Lie group H {\displaystyle H} is an isomorphism of Lie groups if and only if it is bijective.