if $g$ is a lie algebra what is $g^*$?

Question

Iam trying to learn what a coadjoint orbit is but I can't since everywhere I look the definition involves $g^*$.Something that I googled and didn't find anything. I am not even sure what $g^*$. Is it just the dual vector space of our the vector space $V$ of $g$. Is it somehow a lie algebra?, or even maybe even something dual to the notion of lie algebra. Can anyone explain Thanks in advance

Answer 1

A Lie algebra ${\frak g}$ is in particular a vector space, and $\frak{g}^*$ denotes the dual vector space.

It does not come with any canonical Lie algebra structure, we just think of it as a vector space.

Answer 2

Here $\mathfrak{g}$ also denotes the adjoint module, given by the adjoint representation $ad\colon \mathfrak{g}\rightarrow \mathfrak{gl}(\mathfrak{g})$. It is a $\mathfrak{g}$-module. Its dual module is denoted by $\mathfrak{g}^{\ast}$. So I would answer the title question as follows:

"If $\mathbb{g}$ is a Lie algebra - then $\mathfrak{g}$ and $\mathfrak{g}^{\ast}$ are $\mathfrak{g}$-modules".