# Maximal torus

In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.

A torus in a compact Lie group G is a compact, connected, abelian Lie subgroup of G (and therefore isomorphic to the standard torus Tn). A maximal torus is one which is maximal among such subgroups. That is, T is a maximal torus if for any torus T containing T we have T = T. Every torus is contained in a maximal torus simply by dimensional considerations. A noncompact Lie group need not have any nontrivial tori (e.g. Rn).

The dimension of a maximal torus in G is called the rank of G. The rank is well-defined since all maximal tori turn out to be conjugate. For semisimple groups the rank is equal to the number of nodes in the associated Dynkin diagram.

## Examples

The unitary group U(n) has as a maximal torus the subgroup of all diagonal matrices. That is,

$T=\left\{\operatorname {diag} \left(e^{i\theta _{1}},e^{i\theta _{2}},\dots ,e^{i\theta _{n}}\right):\forall j,\theta _{j}\in \mathbb {R} \right\}.$ T is clearly isomorphic to the product of n circles, so the unitary group U(n) has rank n. A maximal torus in the special unitary group SU(n) ⊂ U(n) is just the intersection of T and SU(n) which is a torus of dimension n  1.

A maximal torus in the special orthogonal group SO(2n) is given by the set of all simultaneous rotations in any fixed choice of n pairwise orthogonal planes (i.e., two dimensional vector spaces). Concretely, the maximal torus consists of all block-diagonal matrices with $2\times 2$ diagonal blocks, where each diagonal block is a rotation matrix. This is also a maximal torus in the group SO(2n+1) where the action fixes the remaining direction. Thus both SO(2n) and SO(2n+1) have rank n. For example, in the rotation group SO(3) the maximal tori are given by rotations about a fixed axis.

The symplectic group Sp(n) has rank n. A maximal torus is given by the set of all diagonal matrices whose entries all lie in a fixed complex subalgebra of H.

## Properties

Let G be a compact, connected Lie group and let ${\mathfrak {g}}$ be the Lie algebra of G. The first main result is the torus theorem, which may be formulated as follows:

Torus theorem: If T is one fixed maximal torus in G, then every element of G is conjugate to an element of T.

This theorem has the following consequences:

• All maximal tori in G are conjugate.
• All maximal tori have the same dimension, known as the rank of G.
• A maximal torus in G is a maximal abelian subgroup, but the converse need not hold.
• The maximal tori in G are exactly the Lie subgroups corresponding to the maximal abelian subalgebras of ${\mathfrak {g}}$ (cf. Cartan subalgebra)
• Every element of G lies in some maximal torus; thus, the exponential map for G is surjective.
• If G has dimension n and rank r then n r is even.

## Root system

If T is a maximal torus in a compact Lie group G, one can define a root system as follows. The roots are the weights for the adjoint action of T on the complexified Lie algebra of G. To be more explicit, let ${\mathfrak {t}}$ denote the Lie algebra of T, let ${\mathfrak {g}}$ denote the Lie algebra of $G$ , and let ${\mathfrak {g}}_{\mathbb {C} }:={\mathfrak {g}}\oplus i{\mathfrak {g}}$ denote the complexification of ${\mathfrak {g}}$ . Then we say that an element $\alpha \in {\mathfrak {t}}$ is a root for G relative to T if $\alpha \neq 0$ and there exists a nonzero $X\in {\mathfrak {g}}_{\mathbb {C} }$ such that

$\mathrm {Ad} _{e^{H}}(X)=e^{i\langle \alpha ,H\rangle }X$ for all $H\in {\mathfrak {t}}$ . Here $\langle \cdot ,\cdot \rangle$ is a fixed inner product on ${\mathfrak {g}}$ that is invariant under the adjoint action of connected compact Lie groups.

The root system, as a subset of the Lie algebra ${\mathfrak {t}}$ of T, has all the usual properties of a root system, except that the roots may not span ${\mathfrak {t}}$ . The root system is a key tool in understanding the classification and representation theory of G.

## Weyl group

Given a torus T (not necessarily maximal), the Weyl group of G with respect to T can be defined as the normalizer of T modulo the centralizer of T. That is,

$W(T,G):=N_{G}(T)/C_{G}(T).$ Fix a maximal torus $T=T_{0}$ in G; then the corresponding Weyl group is called the Weyl group of G (it depends up to isomorphism on the choice of T).

The first two major results about the Weyl group are as follows.

• The centralizer of T in G is equal to T, so the Weyl group is equal to N(T)/T.
• The Weyl group is generated by reflections about the roots of the associated Lie algebra. Thus, the Weyl group of T is isomorphic to the Weyl group of the root system of the Lie algebra of G.

We now list some consequences of these main results.

• Two elements in T are conjugate if and only if they are conjugate by an element of W. That is, each conjugacy class of G intersects T in exactly one Weyl orbit. In fact, the space of conjugacy classes in G is homeomorphic to the orbit space T/W.
• The Weyl group acts by (outer) automorphisms on T (and its Lie algebra).
• The identity component of the normalizer of T is also equal to T. The Weyl group is therefore equal to the component group of N(T).
• The Weyl group is finite.

The representation theory of G is essentially determined by T and W.

As an example, consider the case $G=SU(n)$ with $T$ being the diagonal subgroup of $G$ . Then $x\in G$ belongs to $N(T)$ if and only if $x$ maps each standard basis element $e_{i}$ to a multiple of some other standard basis element $e_{j}$ , that is, if and only if $x$ permutes the standard basis elements, up to multiplication by some constants. The Weyl group in this case is then the permutation group on $n$ elements.

## Weyl integral formula

Suppose f is a continuous function on G. Then the integral over G of f with respect to the normalized Haar measure dg may be computed as follows:

$\displaystyle {\int _{G}f(g)\,dg=|W|^{-1}\int _{T}|\Delta (t)|^{2}\int _{G/T}f\left(yty^{-1}\right)\,d[y]\,dt,}$ where $d[y]$ is the normalized volume measure on the quotient manifold $G/T$ and $dt$ is the normalized Haar measure on T. Here Δ is given by the Weyl denominator formula and $|W|$ is the order of the Weyl group. An important special case of this result occurs when f is a class function, that is, a function invariant under conjugation. In that case, we have

$\displaystyle {\int _{G}f(g)\,dg=|W|^{-1}\int _{T}f(t)|\Delta (t)|^{2}\,dt.}$ Consider as an example the case $G=SU(2)$ , with $T$ being the diagonal subgroup. Then the Weyl integral formula for class functions takes the following explicit form:

$\displaystyle {\int _{SU(2)}f(g)\,dg={\frac {1}{2}}\int _{0}^{2\pi }f\left(\mathrm {diag} \left(e^{i\theta },e^{-i\theta }\right)\right)\,4\,\mathrm {sin} ^{2}(\theta )\,{\frac {d\theta }{2\pi }}.}$ Here $|W|=2$ , the normalized Haar measure on $T$ is ${\frac {d\theta }{2\pi }}$ , and $\mathrm {diag} \left(e^{i\theta },e^{-i\theta }\right)$ denotes the diagonal matrix with diagonal entries $e^{i\theta }$ and $e^{-i\theta }$ .

## See also

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