4

I have seen the notion of the abstract Jordan decomposition be given for semisimple Lie algebras over an algebraically closed field of char = 0 in a number of places, a few theorems proved about it (The abstract and usual Jordan decompositions of semisimple linear Lie algebras agree with one another), but I haven't seen what role this concepts plays in understanding semisimple Lie algebras. What I am looking for is what was the purpose of this definition? Can it be used to prove some important theorems of Lie algebras or is it just something amusing that the author decided add?

My knowledge of Lie algebras is limited, I have read the first couple chapters of Humphry's, and I have some other bits and pieces of knowledge from other places, but I am happy to get answers that may be over my head.

john
  • 41

1 Answers1

4

The abstract Jordan decomposition plays a major role for the structure theory of semisimple Lie algebras, beginning with Cartan subalgebras, using the root space decomposition with the Jordan decomposition of $ad(h)$ for $h\in H$. Indeed, the proof of many standard results for semisimple Lie algebras uses the Jordan decomposition. Here is just one example:

Proposition: Let $L$ be a semisimple Lie algebra over an algebraically closed field of characteristic zero. Let $H$ be a Cartan subalgebra of $L$. Then all elements in $H$ are semisimple.

An abstract Jordan decomposition exists for all perfect Lie algebras. As you have mentioned, for linear semisimple Lie algebras the abstract and concrete Jordan decomposition coincides, see this MSE question.

Community
  • 1
Dietrich Burde
  • 130,978