Cartan subalgebra

In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra ${\mathfrak {h}}$ of a Lie algebra ${\mathfrak {g}}$ that is self-normalising (if $[X,Y]\in {\mathfrak {h}}$ for all $X\in {\mathfrak {h}}$ , then $Y\in {\mathfrak {h}}$ ). They were introduced by Élie Cartan in his doctoral thesis.

In a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero, a Cartan subalgebra is the same thing as a maximal toral subalgebra (a toral subalgebra is a subalgebra consisting of semisimple elements). Because of this, a Cartan subalgebra of a semisimple Lie algebra is sometimes defined as a maximal toral subalgebra.

Existence and uniqueness

Cartan subalgebras exist for finite-dimensional Lie algebras whenever the base field is infinite. One way to construct a Cartan subalgebra is by means of a regular element. Over a finite field, the question of the existence is still open.

For a finite-dimensional semisimple Lie algebra ${\mathfrak {g}}$ over an algebraically closed field of characteristic zero, there is a simpler approach: by definition, a toral subalgebra is a subalgebra of ${\mathfrak {g}}$ that consists of semisimple elements (an element is semisimple if the adjoint endomorphism induced by it is diagonalizable). A Cartan subalgebra of ${\mathfrak {g}}$ is then the same thing as a maximal toral subalgebra and the existence of a maximal toral subalgebra is easy to see.

In a finite-dimensional Lie algebra over an algebraically closed field of characteristic zero, all Cartan subalgebras are conjugate under automorphisms of the algebra, and in particular are all isomorphic. The common dimension of a Cartan subalgebra is then called the rank of the algebra.

For a finite-dimensional complex semisimple Lie algebra, the existence of a Cartan subalgebra is much simpler to establish, assuming the existence of a compact real form.[1] In that case, ${\mathfrak {h}}$ may be taken as the complexification of the Lie algebra of a maximal torus of the compact group.

Kac–Moody algebras and generalized Kac–Moody algebras also have Cartan subalgebras.

Cartan subalgebras of semisimple Lie algebras

A Cartan subalgebra of a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic 0 is abelian and also has the following property of its adjoint representation: the weight eigenspaces of ${\mathfrak {g}}$ restricted to ${\mathfrak {h}}$ diagonalize the representation, and the eigenspace of the zero weight vector is ${\mathfrak {h}}$ . (So, the centralizer of ${\mathfrak {h}}$ coincides with ${\mathfrak {h}}$ .) The non-zero weights are called the roots, and the corresponding eigenspaces are called root spaces, and are all 1-dimensional. (See Cartan subalgebras of semisimple Lie algebras for more information.)

If ${\mathfrak {g}}$ 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 ${\mathfrak {g}}$ is the centralizer of a maximal toral subalgebra of ${\mathfrak {g}}$ . If ${\mathfrak {g}}$ 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 ${\mathfrak {g}}$ is semisimple, then the adjoint representation presents ${\mathfrak {g}}$ as a linear Lie algebra, so that a subalgebra of ${\mathfrak {g}}$ is Cartan if and only if it is a maximal toral subalgebra.

Examples

Any nilpotent Lie algebra is its own Cartan subalgebra.
A Cartan subalgebra of the Lie algebra of n×n matrices over a field is the algebra of all diagonal matrices.
The Lie algebra sl₂(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_2n(C) of 2n by 2n matrices of trace 0 has a Cartan subalgebra of rank 2n−1 but has a maximal abelian subalgebra of dimension n² consisting of all matrices of the form ${0\ A \choose 0\ 0}$ 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 (or, since it is normalized by diagonal matrices).

Splitting Cartan subalgebra

Over non-algebraically closed fields, not all Cartan subalgebras are conjugate. An important class are splitting Cartan subalgebras: if a Lie algebra admits a splitting Cartan subalgebra ${\mathfrak {h}}$ then it is called splittable, and the pair $({\mathfrak {g}},{\mathfrak {h}})$ 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.

gollark: You could maybe get away with, somehow, a hardware-based JS parser and interpreter in addition to your premade static compiler thing.

gollark: `eval`?

gollark: Okay, *how* is JS to be compiled to hardware without dropping features?

gollark: I mean, at some point you would just have to shove a JS interpreter on an actual CPU on board.

gollark: It's also practically impossible because JS is too dynamic (`eval`, `new Function`, `fn.toSource()`, the various `call`y methods, whatever is going on with empty slots in arrays, the ability to set properties on arrays too).

References

Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics, 126 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-97370-8, MR 1102012
Jacobson, Nathan (1979), Lie algebras, New York: Dover Publications, ISBN 978-0-486-63832-4, MR 0559927
Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90053-7
Popov, V.L. (2001) [1994], "Cartan subalgebra", Encyclopedia of Mathematics, EMS Press

Hall 2015 Chapter 7

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Hall 2015 Chapter 7