Lie group homomorphism

In mathematics, a Lie group homomorphism is a map between Lie groups

\( \phi\colon G \to H\, \)

which is both a group homomorphism and a smooth map. Lie group homomorphisms are the morphisms in the category of Lie groups. In the case of complex Lie groups, one naturally requires homomorphisms to be holomorphic functions. In either the real or complex case, it is actually sufficient to only require the maps to be continuous. Every continuous homomorphism between Lie groups turns out to be an analytic function.

An isomorphism of Lie groups is a homomorphism whose inverse is also a homomorphism. Equivalently, it is a diffeomorphism which is also a group homomorphism.

Let \( \phi\colon G \to H \) be a Lie group homomorphism and let \( \phi_{*} be its derivative at the identity. If we identify the Lie algebras of G and H with their tangent spaces at the identity then \phi_{*} is a map between the corresponding Lie algebras:

\( \phi_{*}\colon\mathfrak g \to \mathfrak h \)

One can show that \( \phi_{*} \) is actually 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 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.

One of the most important properties of Lie group homomorphisms is that the maps \( \phi \) and \( \phi_{*} \) are related by the exponential map. For all \( x\in\mathfrak g \) we have

\( \phi(\exp(x)) = \exp(\phi_{*}(x)).\, \)