I've started studying Kac-Moody algebras and free lie algebras is a really new thing for me. I am trying to understand the statement (b) of theorem 1.2 in the following book Theorem 1.2, statement (b), more specifically:
from the fact that $\mathfrak{g}$ is the free lie algebra on $v_1,\cdots, v_n$ and that $\varphi$ is onto, why one can conclude that $\mathfrak{\tilde n_-}$ is freely generated by $f_1,\cdots, f_n$.
I have some toughts. Here they are:
It have been proven in the book that the homomorphism $\varphi$ maps $\mathfrak{\tilde n_-}$ onto $\mathfrak g$. Also, we may define $\phi: \{v_1,\cdots,v_n\} \rightarrow \mathfrak{\tilde n_-}$ mapping $v_i\mapsto f_i$. Since $\mathfrak g$ is the free lie algebra on $v_1,\cdots,v_n$, there exists a unique homomorphism $\tilde \phi: \mathfrak g \rightarrow \mathfrak{\tilde n_-}$ extending $\phi$. So $\tilde \phi(v_i)=f_i$ and also $\varphi(f_i)=v_i$ and since both maps are lie homomorphism, we may see that both are inverse of each other. Hence $\mathfrak g$ and $\mathfrak{\tilde n_-}$ are isomorphic.
Is this a good path? How can I conclude that $\mathfrak{\tilde n_-}$ is freely generated by $f_1,\cdots, f_n$? Thanks in advance!
