6
$\begingroup$

I am having trouble with the following Lie Algebra question. I will appreciate any help greatly.

Any Lie group homomorphism $\phi : G \rightarrow H$ is determined by the induced Lie algebra homomorphism $d\phi : \mathfrak{g} \rightarrow \mathfrak{h}.$

$\endgroup$
1
$\begingroup$

The derivative $d\phi$ determines $\phi$ on a neighbourhood of the identity of $G$. The image of $\phi$ on the subset of $G$ covered by one-parameter subgroups can be given explicitly in terms of $d\phi$, using the formula $\phi (\exp(tX)) = \exp (t\, d\phi (X))$ for $X \in \mathfrak{g}$. If $G$ is connected then any element of $G$ can be written as a product of elements in one-parameter subgroups, so in this case $d\phi$ determines $\phi$.

However if $G$ is not connected then $\phi$ is not determined by $d \phi$. For example, if $G$ is a discrete group then $\mathfrak{g} = 0$, so $d \phi$ cannot possibly give any useful information.

$\endgroup$

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.