I just started learning Lie groups and I was a bit confused with the definition. The notes I am reading states "a Lie group $G$ is a group endowed with the structure of a $C^{\infty}$ manifold such that the inverse map and the multiplication map are smooth".
With this definition is it correct to assume that I am only thinking of Lie groups as real manifolds, and the corresponding Lie algebra as a vector space over $\mathbb{R}$? The notes doesn't specify the base field, so I was hoping someone could clarify this...
Thank you.
ps If one wishes to consider a complex version instead, I suppose one can do this by replacing $C^{\infty}$ with holomorphic?
Usually, yes, that means that you are dealing with real manifolds. In the case of complex manifolds, we deal with analytic functions then.