Questions about Lie algebras

Question

I would like to know about the application of semisimple Lie algebras in general. In particular its application in ordinary differential equations and its solution advantages with respect to the classical solution. Are there open problems in Lie algebras ?. A concrete example of an ordinary differential equation solved by semi-simple Lie algebras ?. General references on research in Lie algebras

Answer 1

There are several open problems "in Lie algebras", for semisimple, reductive, complete, solvable, nilpotent Lie algebras, just to name a few cases. Some arise from physics, from geometry and from number theory, but there are several more of course. For a survey see here, for "pre-Lie algebras" and associated Lie algebras.

Further examples: Classify simple Lie algebras over fields of characteristic $p=2$ and $p=3$. For $p\ge 5$ and algebraically closed fields there is a (complicated) classification. For a survey see here.