Maybe I am being stupid in my previous question.
Is the representation theory of Lie groups, and Algebraic groups the same for semisimple or reductive groups? I suppose the definition of a representation doesn't make use of the topology of the space, so they are?
A representation of $G$ consists of a finite-dimensional vector space $V$ over a field $K$ along with a group homomorphism $G → GL(V )$. In the case where G is a Lie group, however, we ask the map $G → GL(V)$ to be a smooth map, and the field $K$ usually to be $\mathbb{R}$ or $\mathbb{C}$. So the topology is involved. For algebraic groups, arbitrary fields are possible. But the analytic techniques used for studying Lie groups must be replaced by techniques from algebraic geometry, where the relatively weak Zariski topology causes many technical complications. So representation theory is very different in this case.