Lie group homomorphism ($F:G\to H,F(ab)=F(a)F(b)$)

A smooth homomorphism of Lie groups has differential $D_{e}F$ at the identity. Differentiating $F\circ L_g=L_{F(g)}\circ F$ makes the corresponding left-invariant vector fields $F$-related. Their commutator action on pulled-back functions shows that the differential of a Lie group homomorphism preserves Lie brackets. The map need not be injective or surjective.

An isomorphism of Lie groups is a group isomorphism that is also a diffeomorphism. Both the homomorphism and its inverse preserve the smooth structures. Its identity differential is a Lie algebra isomorphism, but an isomorphism of Lie algebras determines a global Lie group isomorphism without extra period data only for connected simply connected groups.

A Lie group homomorphism intertwines one-parameter subgroups and therefore obeys the displayed identity. A connected Lie group is generated by an identity neighbourhood in an exponential chart, so its homomorphism is uniquely determined by the differential at the identity. On a disconnected group, only the restriction to its identity component is determined this way; the images of the other components require extra data.