I have a probably basic question about modules over Lie algebras which I can not answer due to my very limited knowledge about the algebraic side of Lie theory. I would be happy if someone directs me to where I should read about that. Let $L$ be a simple Lie algebra (assume base field is $\mathbb{C}$ if it helps). Let $M,N$ be irreducible modules over $L$. What is known about the decomposition of $M\otimes N$ into a direct sum of irredubible modules ?
If the above question can not be answered in full generality, then I am also interesred to know what happens in the special case when $M=N=L$ ?