Definition 17.15.1 (Lie algebra homomorphism). A linear map between Lie algebras is a homomorphism if for all .
Proposition 17.15.1 (Functoriality). Every Lie group homomorphism induces a Lie algebra homomorphism (the differential at the identity). This defines a functor from (connected, simply-connected) Lie groups to Lie algebras, and this functor is an equivalence of categories: a Lie algebra homomorphism integrates to a unique Lie group homomorphism (where is the simply-connected cover of ).
Definition 17.15.2 (Representation of a Lie algebra). A representation of a Lie algebra on a vector space is a Lie algebra homomorphism , i.e., a linear map satisfying . A representation of a Lie group on (a smooth homomorphism ) induces a representation of on by differentiation.
Example 17.15.1 (The adjoint representation). Every Lie algebra has a natural representation on itself: the adjoint representation defined by . The Jacobi identity is precisely the statement that is a Lie algebra homomorphism.