Is it correct to say that a non semisimple representation of a vector space is a representation which is reducible and indecomposable?
If yes, how can this intuition be translated to the definition of a non semisimple Lie Algebra which, if I understand correctly, is a Lie Algebra with a non-trivial abelian ideal?
Best Answer
I think there's a little misunderstanding here. Vector spaces don't have representations (at least not any that I am aware of). Do you mean a representation on a vector space?
More generally, semisimple means that it breaks up as the sum (or product, etc.) of 'simple' things. In the case of representations, simple means irreducible and in the case of Lie algebras simple means having no proper ideals.
These are sort of independent but it so happens that the (finite dimensional) representations of a semisimple Lie algebra are semisimple.
Edit: I said this last bit was true for all reductive Lie algebras but that is incorrect. Reductive means that the adjoint representation specifically is completely reducible (a.k.a. semisimple) while semisimple implies that all representations are.