Lie groups and algebraic groups both correspond with Lie algebras, which are by definition the left invariant vector field. But the topology of Lie groups and algebraic groups are different. Are their Lie algebras different?
More concretely, let $G=GL(n,\mathbb C)$ for $n$ a positive integer. When it is considered as a Lie group, the production is given the product topology. But when it is considered as an algebraic group, it is the Zariski topology that is used. In the first case, $G$ is not connected, while in the second case, $G$ is connected. Both of them have Lie algebra $M(n,\mathbb C)$. Are the two Lie algebras "really" all the same?
If they are all the same, is this always the case when $G$ is an arbitrary group which can be considered both as a Lie group and as an algebraic group?