Mapping between Lie algebras

1. Sep 29, 2011

Ted123

If a mapping between Lie algebras $\varphi : \mathfrak{g} \to \mathfrak{h}$ takes a basis in $\mathfrak{g}$ to a basis in $\mathfrak{h}$ is it an isomorphism of vector spaces?

2. Sep 29, 2011

Office_Shredder

Staff Emeritus
Re: Mapping

Good question. What do you think?

3. Sep 30, 2011

Ted123

Re: Mapping

I'm fairly sure it is. Is that right?

4. Sep 30, 2011

HallsofIvy

Re: Mapping

Assuming that by "takes a basis to a basis" you mean "one to one and onto", a Lie Algebra is completely determined by its basis, isn't it?