Let $G$ be a lie group.

Let $$C: G\times G \longrightarrow G$$ $$C(g,h)=ghg^{-1}$$

I have proven that the left $L_g$ and right $R_g$ multiplications are smooth. Therefore $$C_g: G \longrightarrow G$$ $$C_g(h)=ghg^{-1}$$

But how do I use these facts to prove that $C$ is smooth? I'm trying to write it as the composition of smooth functions but I am having trouble doing so.


1 Answer 1


The map $\phi:G\times G\rightarrow G, (g,h)\mapsto g^{-1}$ is smooth, since it is the projection onto the first factor composed with the inverse map, which are both smooth. Thus $G\times G\rightarrow G\times G, (g,h)\mapsto (gh,g^{-1})$ is smooth, as it is the product of $\phi$ and the multiplication map. Applying the multiplication map again gives the result.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.