For a Lie group, one typically requires the representations to be smooth or admissible.
Maschke's theorem states that when the characteristic of F doesn't divide the order of G, the category of representations of G over F is semisimple.
The basic question is whether the decomposition into irreducible representations (simple objects of the category) behaves under restriction or induction.
Tannaka–Krein duality concerns the interaction of a compact topological group and its category of linear representations.
Krein's theorem in effect completely characterizes all categories that can arise from a group in this fashion.
These concepts can be applied to representations of several different structures, see the main article for details.