A question about Lie group homomorphisms

Question

Suppose I have a Lie group $G$ and a Lie homomorphism $ \phi : G \rightarrow GL_n(\mathbb{R})$.

Can $ \phi $ be viewed as some sort of representation of $G$?

Can anyone make this rigorous for me please? Thanks.

Answer 1

Another common way of defining a representation is that it's a vector space $V$ together with a smooth map $G \times V \to V$ that satisfies the axioms of an action by linear transformations.

First convince yourself that the canonical action $\operatorname{GL}_n(\mathbb R) \times \mathbb R^n \to \mathbb R^n$ is smooth. Then given $\phi$ as above you have the map $$G \times \mathbb R^n \xrightarrow{\phi\times\operatorname{id}}\operatorname{GL}_n(\mathbb R) \times \mathbb R^n \to \mathbb R^n$$ which is a composition of smooth maps, so it's smooth and gives the action that you're looking for.

Answer 2

A group representation is a group homomorphism $\phi:G\rightarrow \operatorname{GL}(V)$, where $V$ is a vector space. If we take $V=\mathbb{R}^n$, we see that $\operatorname{GL}(V)=\operatorname{GL}_n\left(\mathbb{R}\right)$. So, what you have got there is a real representation of finite dimension.