3
$\begingroup$

Let $\mathfrak g$ be a complex semisimple Lie algebra. A theorem of Weyl says that every finite-dimensional representation of $\mathfrak g$ is completely reducible. Another very important theorem is that the common Jordan decomposition of endomorphisms in a complex vector space has an abstract counterpart in abstract semisimple Lie algebras $\mathfrak g$. My question is basically: do we need these two theorems for proving the basic facts about the root space decomposition (i.e. structure theory of semisimple Lie algebras)?

1) One example: in many places, the proof that a Cartan subalgebra $\mathfrak h$ (defined generally as nilpotent algebra that coincides with its normalizer in $\mathfrak g$) of a semisimple Lie algebra consists entirely of elements $X$ such that $ad(X)$ is semisimple - is based on above mentioned abstract Jordan decomposition. But in Dixmier, Enveloping algebras, page 38, Dixmier does not seem to use the abstract Jordan decomposition but only the usual one for endomorphisms (and the weight decomposition of $\mathfrak g$ given by the adjoint representation of $\mathfrak h$ on $\mathfrak g$ and that all derivations of a semisimple Lie algebra are inner).

2) The proof of the abstract Jordan decomposition is in many places (e.g. Humphreys) based on the theorem of Weyl about complete reducibility. But in Kirillov, An Introduction to Lie groups and Lie algebras, chapter 6.4, the proof does not seem to be needing Weyl's theorem.

3) A remark: I know that the fact that all derivations of $\mathfrak g$ are inner can be proved with the theorem of Weyl. But it can also be proved using 'only' the nondegeneracy of the Killing form. So here again we do not need the theorem of Weyl.

4) So summarizing I do not see where we would need these two theorems to show that a Cartan subalgebra of $\mathfrak g$ is abelian and consists of semisimple elements. But given that, do we need at any point of proving the basic properties of the root space decomposition these two theorems? In my understanding, only the classification of irreducible $sl_2$-modules plays a role here. We do not need to know that any finite-dimensional module is a direct sum of irreducible ones or that $\rho(H)$ for $H \in \mathfrak g$ semisimple, is semisimple for any representation $\rho$.

$\endgroup$
3
$\begingroup$

I will try an answer. Let us start with the title question. Yes, the abstract Jordan decomposition is used in the theory of semsimple Lie algebras, see this MSE-question. Weyl's theorem itself seems more important for the representation theory of semisimple Lie algebras. But remark $3$ is true. We could prove that a semisimple Lie algebra $L$ of characteristic zero is complete, i.e., that $Der(L)=ad(L)$ and $Z(L)=0$ by Weyl's theorem. Algebraically, however, we would probably replace Weyl's theorem with Whitehead's first Lemma, that $H^1(L,M)=0$ for all finite-dimensional $L$-modules $M$, and $L$ semisimple. For the adjoint module $M=L$ we obtain then $Der(L)=ad(L)$.

Summarizing, I would think, both theorems are fundamental, but there is no need to worry about whether or not they are needed for the structure theory. Why should we want not to use such fundamental results, even if it were possible?

$\endgroup$
  • $\begingroup$ In point 1 I claim that Dixmier does only use the 'concrete' Jordan decomposition to prove that every element in a CSA is semisimple. While I totally agree that both theorems are fundamental in the general representation theory of semisimple Lie algebras I am interest if it is possible to postpone a proof of both to after one proved the basic facts about the root space decomposition. $\endgroup$ – Mekanik Nov 8 '16 at 0:27
  • $\begingroup$ Yes, I think in principle it would be possible to postpone the proofs and use seemingly different arguments. However, in the end we might discover, that these arguments more or less are in the spirit of those two fundamental results, but we would no longer recognize this so clearly. Would this be of any advantage? $\endgroup$ – Dietrich Burde Nov 8 '16 at 10:00
  • $\begingroup$ I think it is in general desirable to know different paths to a result and also to know when one result is indispensable for another. I think I realized one important point where one needs to use the existence of the abstract Jordan decomposition and this is the proof that Cartan subalgebras actually exist. But as I still think that the Jordan decomposition does not need the theorem of Weyl I think one can in fact postpone the proof of Weyl to after one has done the root space structure theory and starts with representation theory in general. $\endgroup$ – Mekanik Nov 8 '16 at 22:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.