Why do we have to classify semisimple Lie algebras?

Question

Almost all Lie algebras textbooks deal with the classification of semisimple Lie algebras. Why is it so important?

Answer 1

Actually, in any topic it is an interesting question to classify (i.e. determine the complete list of isomorphism classes of) the objects you investigate.

In some topics, this task is quite easy (and the classifier is so simple that one doesn't even consider it as such); for example, the classification of vector spaces over a field $k$ is dealt with by the concept of dimension: Two vector spaces are isomorphic if they have the same dimension and for each cardinality there exists a vector space of that dimension. Knowing this classification you can simplify your work, for example in order to prove a statement about finite-dimensional vector spaces - just prove it for $k^n$, $n\in\mathbb N_0$.

In other topics, classification is just slightly harder, for example consider the classification of finite (or finitely generated) abelian groups. In still other topics, the classification is a really tough task, such as in the classification of finite simple groups; such is of course beyond the scope of any normal course.

Anyway, in the light of the above it should not be surprising that one wants to classify semisimple Lie algebras - and fortunately this classification is a doable task and at the same quite rewarding due to the techniques employed and also equips you with a handful of interesting special cases to look out for (think Milnor sphere) in any of the many applications.