|
|
| Line 1: |
Line 1: |
| {{Lie groups |Semi-simple}}
| | <br><br>Luke is [http://lukebryantickets.flicense.com buy luke bryan tickets] actually a celebrity during the making plus the occupation development very first 2nd to his 3rd place record, And , will be the resistant. He burst to the picture in 2008 together with his best combination of [http://lukebryantickets.pyhgy.com luke bryan tour date] lower-property ease of access, movie celebrity great appearance and [http://lukebryantickets.iczmpbangladesh.org cheap tickets for luke bryan] lyrics, is scheduled t in the main way. The latest recor on the country graph or chart and #2 in the put maps, generating it the next top very first in those days of 2013 to get a region designer. <br><br>The child of your , is aware of persistence and willpower are important elements in relation to an excellent career- . His very first album, Continue to be Me, produced the [http://www.hotelsedinburgh.org last minute tickets] Top reaches “All My Pals Say” and “Country Man,” when his hard work, Doin’ Issue, found the vocalist-three direct No. 8 singles: Different Calling Can be a Excellent Point.”<br><br>During the slip of 2006, Concert tour: Luke & which in fact had an amazing set of , which includes City. “It’s much like you are receiving a approval to travel to another level, says individuals artists that were a part of the Concertsover in a bigger measure of artists.” It twisted as the best trips in its ten-12 months background.<br><br>Here is my blog post: concert dates for luke bryan ([http://lukebryantickets.lazintechnologies.com lukebryantickets.lazintechnologies.com]) |
| | |
| In [[mathematics]], a '''Cartan subalgebra''', often abbreviated as '''CSA''', is a [[nilpotent Lie algebra|nilpotent]] [[subalgebra]] <math>\mathfrak{h}</math> of a [[Lie algebra]] <math>\mathfrak{g}</math> that is [[self-normalising]] (if <math>[X,Y] \in \mathfrak{h}</math> for all <math>X \in \mathfrak{h}</math>, then <math>Y \in \mathfrak{h}</math>). They were introduced by [[Élie Cartan]] in his doctoral thesis.
| |
| | |
| == Existence and uniqueness ==
| |
| | |
| Cartan subalgebras exist for finite dimensional Lie algebras whenever the base field is infinite. If the field is algebraically closed of characteristic 0 and the algebra is finite dimensional then all Cartan subalgebras are conjugate under automorphisms of the Lie algebra, and in particular are all isomorphic.
| |
| | |
| [[Kac–Moody algebra]]s and [[generalized Kac–Moody algebra]]s also have Cartan subalgebras.
| |
| | |
| == Properties ==
| |
| | |
| A Cartan subalgebra of a finite dimensional [[semisimple Lie algebra]] over an [[algebraically closed field]] of characteristic 0 is [[abelian Lie algebra|abelian]] and
| |
| also has the following property of its [[adjoint representation of a Lie algebra|adjoint representation]]: the [[weight (representation theory)|weight]] [[eigenspace]]s of <math>\mathfrak{g}</math> [[restricted representation|restricted]] to <math>\mathfrak{h}</math> diagonalize the representation, and the eigenspace of the zero weight vector is <math>\mathfrak{h}</math>. (So, the centralizer of <math>\mathfrak{h}</math> coincides with <math>\mathfrak{h}</math>.) The non-zero weights are called the '''[[root system|roots]]''', and the corresponding eigenspaces are called '''[[root system|root spaces]]''', and are all 1-dimensional.
| |
| | |
| If <math>\mathfrak{g}</math> is a [[linear Lie algebra]] (a Lie subalgebra of the Lie algebra of endomorphisms of a finite-dimensional vector space ''V'') over an algebraically closed field, then any Cartan subalgebra of <math>\mathfrak{g}</math> is the [[centralizer]] of a maximal [[toral Lie algebra|toral Lie subalgebra]] of <math>\mathfrak{g}</math>; that is, a subalgebra consisting entirely of elements which are diagonalizable as endomorphisms of ''V'' which is maximal in the sense that it is not properly included in any other such subalgebra. If <math>\mathfrak{g}</math> is semisimple and the field has characteristic zero, then a maximal toral subalgebra is self-normalizing, and so is equal to the associated Cartan subalgebra. If in addition <math>\mathfrak g</math> is semisimple, then the [[Adjoint representation of a Lie group|adjoint representation]] presents <math>\mathfrak g</math> as a linear Lie algebra, so that a subalgebra of <math>\mathfrak g</math> is Cartan if and only if it is a maximal toral subalgebra. An advantage of this approach is that it is trivial to show the existence of such a subalgebra. In fact, if <math>\mathfrak g</math> has only nilpotent elements, then it is [[nilpotent Lie algebra|nilpotent]] ([[Engel's theorem]]), but then its [[Killing form]] is identically zero, contradicting semisimplicity. Hence, <math>\mathfrak g</math> must have a nonzero semisimple element.
| |
| | |
| == Examples ==
| |
| | |
| *Any nilpotent Lie algebra is its own Cartan subalgebra.
| |
| *A Cartan subalgebra of the Lie algebra of [[square matrix|''n''×''n'' matrices]] over a field is the algebra of all diagonal matrices.
| |
| *The Lie algebra sl<sub>2</sub>('''R''') of 2 by 2 matrices of trace 0 has two non-conjugate Cartan subalgebras.
| |
| *The dimension of a Cartan subalgebra is not in general the maximal dimension of an abelian subalgebra, even for complex simple Lie algebras. For example, the Lie algebra ''sl''<sub>2''n''</sub>('''C''') of 2''n'' by 2''n'' matrices of trace 0 has a Cartan subalgebra of rank 2''n''−1 but has a maximal abelian subalgebra of dimension ''n''<sup>2</sup> consisting of all matrices of the form <math>{0\ A\choose 0\ 0}</math> with ''A'' any ''n'' by ''n'' matrix. One can directly see this abelian subalgebra is not a Cartan subalgebra, since it is contained in the nilpotent algebra of strictly upper triangular matrices (which is also not a Cartan subalgebra since it is normalized by diagonal matrices).
| |
| | |
| == Splitting Cartan subalgebra ==
| |
| {{main|Splitting Cartan subalgebra}}
| |
| | |
| Over non-algebraically closed fields, not all Cartan subalgebras are conjugate. An important class are [[splitting Cartan subalgebra]]s: if a Lie algebra admits a splitting Cartan subalgebra <math>\mathfrak{h}</math> then it is called ''splittable,'' and the pair <math>(\mathfrak{g},\mathfrak{h})</math> is called a [[split Lie algebra]]; over an algebraically closed field every semisimple Lie algebra is splittable. Any two splitting Cartan algebras are conjugate, and they fulfill a similar function to Cartan algebras in semisimple Lie algebras over algebraically closed fields, so split semisimple Lie algebras (indeed, split reductive Lie algebras) share many properties with semisimple Lie algebras over algebraically closed fields.
| |
| | |
| Over a non-algebraically closed field not every semisimple Lie algebra is splittable, however.
| |
| | |
| == See also ==
| |
| *[[Cartan subgroup]]
| |
| *[[Carter subgroup]]
| |
| | |
| == References ==
| |
| *{{Citation | last1=Borel | first1=Armand | author1-link=Armand Borel | title=Linear algebraic groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-97370-8 | mr=1102012 | year=1991 | volume=126}}
| |
| *{{Citation | last1=Jacobson | first1=Nathan | author1-link=Nathan Jacobson | title=Lie algebras | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-63832-4 | mr=559927 | year=1979}}
| |
| *{{Citation | last1=Humphreys | first1=James E. | title=Introduction to Lie Algebras and Representation Theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90053-7 | year=1972}}
| |
| *{{springer|id=C/c020550|first=V.L.|last= Popov|authorlink=Vladimir L. Popov}}
| |
| | |
| [[Category:Lie algebras]]
| |
Luke is buy luke bryan tickets actually a celebrity during the making plus the occupation development very first 2nd to his 3rd place record, And , will be the resistant. He burst to the picture in 2008 together with his best combination of luke bryan tour date lower-property ease of access, movie celebrity great appearance and cheap tickets for luke bryan lyrics, is scheduled t in the main way. The latest recor on the country graph or chart and #2 in the put maps, generating it the next top very first in those days of 2013 to get a region designer.
The child of your , is aware of persistence and willpower are important elements in relation to an excellent career- . His very first album, Continue to be Me, produced the last minute tickets Top reaches “All My Pals Say” and “Country Man,” when his hard work, Doin’ Issue, found the vocalist-three direct No. 8 singles: Different Calling Can be a Excellent Point.”
During the slip of 2006, Concert tour: Luke & which in fact had an amazing set of , which includes City. “It’s much like you are receiving a approval to travel to another level, says individuals artists that were a part of the Concertsover in a bigger measure of artists.” It twisted as the best trips in its ten-12 months background.
Here is my blog post: concert dates for luke bryan (lukebryantickets.lazintechnologies.com)