Lie Algebras of Covering of a Group is Isomorphic to the Lie Algebra of the Group.

Question

If $\tilde{G} \to G$ is a covering of the lie group $G$, why are the associated lie algebras isomorphic? I.e., why $Lie(\tilde{G}) \cong Lie( G)$?

Answer 1

There is natural identification of $T_eG\cong \mathfrak{g}$, and we know that a covering map of groups $\pi:\widetilde{G}\to G$ is a diffeomorphism in a neighborhood $U$ of the identity, whence it follows that $d\pi_*:T_e\widetilde{G}\to T_eG$ is an isomorphism. In particular, we have an isomorphism $\widetilde{\mathfrak{g}}\to \mathfrak{g}$ of the associated Lie algebras.