Cartan subalgebra

Examples

• Any nilpotent Lie algebra is its own Cartan subalgebra.
• The algebra of all diagonal matrices is a Cartan subalgebra of ${\displaystyle {\mathfrak{g}\mathfrak{l}}_n}$, the Lie algebra of ${\displaystyle n\times n}$ matrices over a field.
• For the special Lie algebra of traceless ${\displaystyle n\times n}$ matrices ${\displaystyle {\mathfrak{s}\mathfrak{l}}_n(\mathbb{C})}$, it has the Cartan subalgebra:$${\displaystyle \mathfrak{h}=\left\{d(a_1,\dots ,a_n)\mid a_i\in \mathbb{C}\text{ and }\sum _{i=1}^n{a}_i=0\right\}}$$ where $${\displaystyle d(a_1,\dots ,a_n)=\left(\begin{array}{cccc}{a}_1& 0& \cdots & 0\\ 0& \ddots & & 0\\ ⋮& & \ddots & ⋮\\ 0& \cdots & \cdots & a_n\end{array}\right)}$$ For example, in ${\displaystyle {\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})}$ the Cartan subalgebra is the subalgebra of matrices:$${\displaystyle \mathfrak{h}=\left\{\left(\begin{array}{cc}a& 0\\ 0& -a\end{array}\right):a\in \mathbb{C}\right\}}$$ with Lie bracket given by the matrix commutator.
• The Lie algebra ${\displaystyle {\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})}$ of ${\displaystyle 2}$ by ${\displaystyle 2}$ matrices of trace ${\displaystyle 0}$ 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 ${\displaystyle {\mathfrak{s}\mathfrak{l}}_{2n}(\mathbb{C})}$of ${\displaystyle 2n}$ by ${\displaystyle 2n}$ matrices of trace ${\displaystyle 0}$ has a Cartan subalgebra of rank ${\displaystyle 2n-1}$ but has a maximal abelian subalgebra of dimension ${\displaystyle n^2}$ consisting of all matrices of the form ${\displaystyle \left(\begin{array}{cc}0& A\\ 0& 0\end{array}\right)}$ with ${\displaystyle A}$ any ${\displaystyle n}$ by ${\displaystyle n}$ 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).

Cartan subalgebras of semisimple Lie algebras

For finite-dimensional semisimple Lie algebra ${\displaystyle \mathfrak{g}}$ over an algebraically closed field of characteristic 0, a Cartan subalgebra ${\displaystyle \mathfrak{h}}$ has the following properties:

• ${\displaystyle \mathfrak{h}}$ is abelian,
• For the adjoint representation ${\displaystyle \mathrm{ad}:\mathfrak{g}\to \mathfrak{g}\mathfrak{l}(\mathfrak{g})}$, the image ${\displaystyle \mathrm{ad}(\mathfrak{h})}$ consists of semisimple operators (i.e., diagonalizable matrices).

(As noted earlier, a Cartan subalgebra can in fact be characterized as a subalgebra that is maximal among those having the above two properties.)

These two properties say that the operators in ${\displaystyle \mathrm{ad}(\mathfrak{h})}$ are simultaneously diagonalizable and that there is a direct sum decomposition of ${\displaystyle \mathfrak{g}}$ as:

${\displaystyle \mathfrak{g}=\underset{\lambda \in {\mathfrak{h}}^{\ast }}{⨁}{\mathfrak{g}}_{\lambda }}$

where

${\displaystyle {\mathfrak{g}}_{\lambda }=\{x\in \mathfrak{g}:\text{ad}(h)x=\lambda (h)x,\text{ for }h\in \mathfrak{h}\}}$.

Let ${\displaystyle \mathrm{\Phi }=\{\lambda \in {\mathfrak{h}}^{\ast }\setminus \{0\}|{\mathfrak{g}}_{\lambda }\ne \{0\}\}}$. Then ${\displaystyle \mathrm{\Phi }}$ is a root system and, moreover, ${\displaystyle {\mathfrak{g}}_0=\mathfrak{h}}$; i.e., the centralizer of ${\displaystyle \mathfrak{h}}$ coincides with ${\displaystyle \mathfrak{h}}$. The above decomposition can then be written as:

${\displaystyle \mathfrak{g}=\mathfrak{h}\oplus \left(\underset{\lambda \in \mathrm{\Phi }}{⨁}{\mathfrak{g}}_{\lambda }\right)}$

As it turns out, for each ${\displaystyle \lambda \in \mathrm{\Phi }}$, ${\displaystyle {\mathfrak{g}}_{\lambda }}$ has dimension one and so:

${\displaystyle \dim \mathfrak{g}=\dim \mathfrak{h}+\mathrm{\#}\mathrm{\Phi }}$.

See also Semisimple Lie algebra#Structure for further information.

Decomposing representations with dual Cartan subalgebra

Given a Lie algebra ${\displaystyle \mathfrak{g}}$ over a field of characteristic ${\displaystyle 0}$, and a Lie algebra representation$${\displaystyle \sigma :\mathfrak{g}\to \mathfrak{g}\mathfrak{l}(V)}$$ there is a decomposition related to the decomposition of the Lie algebra from its Cartan subalgebra. If we set $${\displaystyle V_{\lambda }=\{v\in V:(\sigma (h))(v)=\lambda (h)v\text{ for }h\in \mathfrak{h}\}}$$ with ${\displaystyle \lambda \in {\mathfrak{h}}^{\ast }}$, called the weight space for weight ${\displaystyle \lambda }$, there is a decomposition of the representation in terms of these weight spaces $${\displaystyle V=\underset{\lambda \in {\mathfrak{h}}^{\ast }}{⨁}{V}_{\lambda }}$$ In addition, whenever ${\displaystyle V_{\lambda }\ne \{0\}}$ we call ${\displaystyle \lambda }$ a weight of the ${\displaystyle \mathfrak{g}}$-representation ${\displaystyle V}$.

Classification of irreducible representations using weights

But, it turns out these weights can be used to classify the irreducible representations of the Lie algebra ${\displaystyle \mathfrak{g}}$. For a finite dimensional irreducible ${\displaystyle \mathfrak{g}}$-representation ${\displaystyle V}$, there exists a unique weight ${\displaystyle \lambda \in \mathrm{\Phi }}$ with respect to a partial ordering on ${\displaystyle {\mathfrak{h}}^{\ast }}$. Moreover, given a ${\displaystyle \lambda \in \mathrm{\Phi }}$ such that ${\displaystyle ⟨\alpha ,\lambda ⟩\in \mathbb{N}}$ for every positive root ${\displaystyle \alpha \in {\mathrm{\Phi }}^{+}}$, there exists a unique irreducible representation ${\displaystyle L^{+}(\lambda )}$. This means the root system ${\displaystyle \mathrm{\Phi }}$ contains all information about the representation theory of ${\displaystyle \mathfrak{g}}$.^{pg 240}

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 ${\displaystyle \mathfrak{h}}$ then it is called splittable, and the pair ${\displaystyle (\mathfrak{g},\mathfrak{h})}$ 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.

Cartan subgroup

The Lie group–Lie algebra correspondence allows one to translate the notion of Cartan subalgebras to Lie groups. Specifically, a subgroup ${\displaystyle G\subset H}$ of a Lie Group, translates to a sub Lie Algebra ${\displaystyle Lie(G)\subset Lie(H)}$.

In the case of compact connected Lie groups, cartan subgroups are maximal connected Abelian subgroups and are oftentimes referred to as ‘ maximal tori.’

In the case of disconnected compact lie groups, there are two common definitions. The large Cartan subgroup is defined as the group of elements that normalize a fixed maximal torus and fix the fundamental Weyl chamber. The small Cartan subgroup is defined as the centralizer of a maximal torus.

For any of these definitions, a cartan subgroup of a Lie group implies a Cartan subalgebra relation for the corresponding Lie algebras.

Examples of Cartan Subgroups

• The subgroup in GL₂(R) consisting of diagonal matrices.