This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
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)
Changed ``eigenspace of the zero weight vector is itself" to ``the centralizer of ", 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).
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)
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)
Here's some concrete ways the article could be improved
There is an analogous object in the theory of linear algebraic groups, the character group
of the Maximal torus for a linear algebraic group . This structure is used analogously to the Cartan subalgebra for constructing irreducible representations of . Reference in the D-modules book starting on page 244
This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
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)
Changed ``eigenspace of the zero weight vector is itself" to ``the centralizer of ", 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).
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)
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)
Here's some concrete ways the article could be improved
There is an analogous object in the theory of linear algebraic groups, the character group
of the Maximal torus for a linear algebraic group . This structure is used analogously to the Cartan subalgebra for constructing irreducible representations of . Reference in the D-modules book starting on page 244