0

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 $M=N=L$ ?

Amr
  • 20,030
  • 1
    See also this post. For $\mathfrak{sl}_n(K)$ see here, for example. There are many more posts. – Dietrich Burde Apr 17 '23 at 11:33
  • 1
    To add some more about your specific mentioned example, we should first note that $M\otimes M$ will always decompose into $S^2M$ and $\bigwedge^2M$ (symmetric and antisymmetric pieces) although these will then decompose further in general. Even more specifically, there must be a single copy of the trivial representation in $L\otimes L$ corresponding to the Killing form and this will be in $S^2L$. – Callum Apr 17 '23 at 13:00
  • I have recalled that there should also be a copy of $L$ itself in $\bigwedge^2L$. This is part of a more general fact that a trivial representation in the symmetric/antisymmetric part of $M\otimes M$ gives rise to a copy of the adjoint rep in the other part. I cannot remember a proof of that fact but I believe it is true. More generally though I'm not sure there is a simple pattern. You can compute examples with a program like LiE though if you need. – Callum Apr 17 '23 at 21:08

0 Answers0