Lie group homomorphism

A smooth map between Lie groups that preserves multiplication (and hence inverses).

Lie group homomorphism

Let $G$ and $H$ be Lie groups . A Lie group homomorphism is a map $\varphi:G\to H$ that is both a group homomorphism and a smooth map ; equivalently,

\varphi(gh)=\varphi(g)\varphi(h)\quad \text{for all }g,h\in G, \qquad\text{and}\qquad \varphi \text{ is smooth.}

In particular, $\varphi(e_G)=e_H$ and $\varphi(g^{-1})=\varphi(g)^{-1}$ .

A Lie group homomorphism induces a canonical morphism between Lie algebras: its differential at the identity

(d\varphi)_e:T_eG\to T_{e}H

is a Lie algebra homomorphism, as in the induced map on Lie algebras . It also intertwines the exponential maps :

\varphi(\exp_G X) = \exp_H\big((d\varphi)_e X\big) \qquad\text{for all }X\in T_eG.

The kernel $\ker(\varphi)$ is a subgroup of $G$ ; when it is closed (as happens, for example, for many standard homomorphisms), it is an embedded Lie subgroup of $G$ .

Examples

The determinant
$\det:GL(n,\mathbb{R})\to \mathbb{R}^\times$
is a Lie group homomorphism (multiplicative and smooth).
The map $\exp:\mathbb{R}\to \mathbb{R}_{>0}$ given by $t\mapsto e^t$ is a Lie group homomorphism from $(\mathbb{R},+)$ to $(\mathbb{R}_{>0},\times)$ .
The covering map $\mathbb{R}\to S^1$ given by $t\mapsto e^{it}$ is a Lie group homomorphism from $(\mathbb{R},+)$ onto the circle group.