In the notes I am reading, which is about algebraic groups, in the section about over $\mathbb{R}$ all the sudden they started using the word Lie groups. I understand Lie groups and algebraic groups are related but I am finding this section of the notes confusing.
If I have $G$ a linear algebraic group defined over $\mathbb{Q}$ say, and if I consider $G(\mathbb{R})$ the real points, does it automatically become a Lie group (with the Zariski topology)?
If $G$ is a linear algebraic group over $\mathbb{R}$, (it is automatically smooth and) $G(\mathbb{R})$ can be viewed as a polynomial subset of some affine space $\mathbb{R}^n$. For the usual topology on $\mathbb{R}^n$ (and not the Zariski topology), $G(\mathbb{R})$ is a smooth sub-manifold, and since the multiplication and inverse on $G(\mathbb{R})$ are given by polynomials, they are smooth, so $G(\mathbb{R})$ is a real Lie group in the sens of differential geometry. But I think there are real Lie groups that don't come this way.
It is the same thing for complex algebraic linear groups.