Talk:Cartan subalgebra

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)

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)

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)

Clarification on conjugates

The intro of the article states "all Cartan subalgebras are conjugate under automorphisms of the algebra", while a later example states "..2 by 2 matrices of trace 0 has two **non-conjugate** Cartan subalgebras." There is perhaps a subtle distinction here, can anyone clarify this? — Preceding unsigned comment added by Dzackgarza (talk • contribs) 00:12, 25 January 2020 (UTC)

It’s simple: the real field $\mathbb {R}$ is not algebraically closed. —- Taku (talk) 08:14, 25 January 2020 (UTC)

Could you add this example with the two non-conjugate subalgebras? Upon inspection, I'm guessing these are given by ${\text{Span}}_{\mathbb {R} }(e_{12})$ and ${\text{Span}}_{\mathbb {R} }(e_{21})$, but we should see why these aren't conjugate by writing an explicit conjugation. I'm guessing this will give a quadratic which has a complex root, hence these two subalgebras cannot be conjugate. Wundzer (talk) 16:48, 7 October 2020 (UTC)

Article improvements

Here's some concrete ways the article could be improved

Replacement in theory of algebraic groups

There is an analogous object in the theory of linear algebraic groups, the character group

$L={\text{Hom}}(H,k^{\times })$

of the Maximal torus $H$ for a linear algebraic group $G$. This structure is used analogously to the Cartan subalgebra for constructing irreducible representations of $G$. Reference in the D-modules book starting on page 244

Is it a Cartan subgroup? I agree this article seems like a good place to discuss a Cartan subgroup. -- Taku (talk) 01:25, 7 October 2020 (UTC)

@TakuyaMurata:Ah, yes. It appears that's the case. Maybe we can have a section/subsection mentioning this and write up the theory on the Cartan subgroup page. That way this page doesn't get too bloated/have too many seemingly redundant pieces of information, since the decompositions are very similar in both categories. Wundzer (talk) 16:44, 7 October 2020 (UTC)

I personally think we don't need a separate page for a Cartan subgroup, since at least for complex Lie groups, Cartan subgroups are those whose Lie algebras ere Cartan subalgebras. -- Taku (talk) 02:54, 9 October 2020 (UTC)

Kac-Moody algebras

• Add Cartan subalgebras for Kac-Moody algebras

I disagree that this is a good place to discuss Cartan subalgebras of Kac-Moody algebras. Yes, by definition, there is a subalgebra called a Cartan subalgebra but as far as I understand, it is unrelated to Cartan subalgebras of a general Lie algebra. The terminology choice is unfortunate but that's not my fault :( -- Taku (talk) 01:25, 7 October 2020 (UTC)

I'm not sure I 100% agree with this analysis. Check out page 22 of https://web.archive.org/web/20201006224222/https://www.maths.ed.ac.uk/~igordon/LA1.pdf The formal properties of the Cartan subalgebra defined there are strikingly similar to the Cartan subalgebra defined here. If you look in the D-modules book, they mention the exact same kind of decomposition ${\mathfrak {g}}={\mathfrak {h}}\oplus {\mathfrak {n}}^{+}\oplus {\mathfrak {n}}^{-}$ for Lie algebras. Wundzer (talk) 16:42, 7 October 2020 (UTC)

My point was that the decomposition like that is more to do with the fact that a Cartan subalgebra is a toral subalgebra; i.e., the decomposition is a feature of a toral subalgebra rather than a Cartan subalgebra (in the semisimple case, a Cartan subalgebra happens to coincide with a maximal toral subalgebra). This is why I said the terminology seems problematic and in fact, as I noticed, V. Kac, Infinite-dimensional Lie algebras does not use the term "Cartan subalgebra" at all. Also, Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory also prefers to use the term "maximal toral subalgebra" than a Cartan subalgebra (since, terminologically speaking, that's more correct thing to do) -- Taku (talk) 02:54, 9 October 2020 (UTC)

Decomposition

• There should be another article discussing the decomposition theorem of ${\mathfrak {g}}$ and its relations with other parts of the theory. This could include material from 230-260 in the D-modules book
• Also add a main article reference in this article
• Discuss how a torus can be constructed from the cartan subalgebra.

Decomposition of representations

• Add required material for stating theorem on page 240
• Add example of representations of sl_2
• Add example for sl_3?

Shouldn't this type of the discussion belong to semisimple Lie algebra? (cf. semisimple Lie algebra#Structure and semisimple Lie algebra#Representation theory of semisimple Lie algebras) -- Taku (talk) 01:25, 7 October 2020 (UTC)

Sure, but maybe we should include sl_2 here just for the sake of giving an example within the article. That way the example isn't too cumbersome but still gives the general idea. Thoughts? How else could we direct Wikipedia folks to the correct section of that page? Wundzer (talk) 16:46, 7 October 2020 (UTC)

It makes sense to have some mention of applications of Cartan subalgebras. A Cartan subalgebra certainly plays a central role in the structure theory of a semisimple Lie algebra but the discussion of that should appear, obviously, in semisimple Lie algebra not here; ditto for the representation theory of semisimple Lie algebras. -- Taku (talk) 02:54, 9 October 2020 (UTC)