Talk:Cartan subalgebra

From Wikipedia, the free encyclopedia

WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: Start Class Mid Priority  Field: Algebra

[edit] Full definition given

Added the ``full" definition including the nilpotency property. In the semisimple case the df is as previous versions. Not too sure if too much more can be added without encroaching into other aspects already covered in existing pages on Lie algebras. Dmaher 09:56, 1 June 2006 (UTC)

[edit] Zero weight corrections

Changed ``eigenspace of the zero weight vector is \mathfrak{h} itself" to ``the centralizer of \mathfrak{h}", as this is more generally true (See Humphreys 35). The two don't always coincide (though they do in the cases Humphreys considers, when char F = 0).

The sentence was about the case of algebrically closed fields of characteristic 0 (though this condition was only given somewhat ambiguously in the previous paragraph). I have added the condition explicitly to make things clear. (Without this condition the sentence is completely wrong even with your correction.) R.e.b. 18:05, 12 June 2006 (UTC)

[edit] Maximal abelian subalgebras

R.e.b. just added a nice example showing maximal abelian subalgebras need not be Cartan subalgebras, but the explanation is rather indirect (cartan subalgebras have a unique dimension, the dimension of this abelian subalgebra is bigger, so the maximal abelian subalgebra containing it is an example). I tried to make it a little more explicit by mentioning the identity matrix was not included, but I think my explanation only shows there is a bigger abelian subalgebra. Can someone sharpen the example to show directly a maximal abelian subalgebra which is not self-normalizing? JackSchmidt (talk) 18:24, 11 February 2008 (UTC)

R.e.b. fixed this, thanks! JackSchmidt (talk) 23:28, 11 February 2008 (UTC)