nLab maximal torus

Definition

Let $G$ be a compact Lie group.

A torus in $G$ is an abelian subgroup $T \hookrightarrow G$ which is connected (and compact). Note that any compact connected abelian Lie group must be a torus $U(1)^{\times n}$, hence the name.

A maximal torus is a subgroup which is maximal with these properties.

The Lie algebra of a maximal torus of $G$ is a Cartan subalgebra of the Lie algebra of $G$.

Properties

Suppose throughout that the compact Lie group $G$ is connected.

Proposition

Given a choice of maximal torus $T \hookrightarrow G$, then each element $g \in G$ is conjugate to an element of its maximal torus, i.e. there exists $h_g \in G$ such that $h_g g h_g^{-1} \in T \hookrightarrow G$.

(e.g. Johansen, theorem 2.7.3)

Remark

For the unitary group $U(n)$, by example , prop. is the spectral theorem that unitary matrices may be diagonalized.

Corollary

Any two choices of maximal tori are conjugate.

(e.g. Johansen, corollary 2.7.5)

Proposition

Any two elements of a maximal torus $T\hookrightarrow G$ are conjugate in $G$ precisely if they are conjugate via the Weyl group.

(e.g. Johansen, prop 2.7.13)

Corollary

The conjugacy classes of $G$ are in bijection to $T/W(G,T)$.

Proposition

Under the assumptions on $G$, the exponential map $\exp: \mathfrak{g} \to G$ is surjective.

Proof

The exponential map $\mathbb{R} \to S^1$ is surjective; similarly, if $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$, then the exponential map $\mathfrak{g} \to G$ maps $\mathfrak{h}$ homomorphically onto a maximal torus. Also the exponential map $\mathfrak{g} \to G$ preserves conjugation by any element of $G$, i.e., $\exp(g x g^{-1}) = g\exp(x)g^{-1}$. Then surjectivity follows from Proposition .

Examples

Example

The maximal torus of a unitary group $U(n)$ is the subgroup $U(1)^n\hookrightarrow U(n)$ of diagonal matrices with unitary entries.

Example

The maximal torus of the special unitary group $SU(n+1)$ is again $U(1)^n$ (one dimension lower) included as the subgroup of diagonal matrices whose first $n$ diagonal entries are arbitrary elements of $U(1)$ and whose last diagonal entry is the inverse of their product.

Example

The maximal torus of the special orthogonal group $SO(2n)$ and $SO(2n+1)$ is $U(1)^n \simeq SO(2)^n$ included block-diagonal (with, in the odd-dimensional case, the remaining block entry being the unit).

Related concepts

• maximal subgroup

• maximal compact subgroup

• Cartan subalgebra

• weight (in representation theory), root (in representation theory)

• Weyl group

• orbit method

References

e.g.

• Troels Roussauc Johansen, section 2.6 of Character Theory for Finite Groups and Compact Lie Groups pdf