Maximal torus

Properties

Let G be a compact, connected Lie group and let ${\displaystyle \mathfrak{g}}$ 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 ${\displaystyle \mathfrak{g}}$ (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 ${\displaystyle \mathfrak{t}}$ denote the Lie algebra of T, let ${\displaystyle \mathfrak{g}}$ denote the Lie algebra of ${\displaystyle G}$, and let ${\displaystyle {\mathfrak{g}}_{\mathbb{C}}:=\mathfrak{g}\oplus i\mathfrak{g}}$ denote the complexification of ${\displaystyle \mathfrak{g}}$. Then we say that an element ${\displaystyle \alpha \in \mathfrak{t}}$ is a root for G relative to T if ${\displaystyle \alpha \ne 0}$ and there exists a nonzero ${\displaystyle X\in {\mathfrak{g}}_{\mathbb{C}}}$ such that

${\displaystyle {\mathrm{A}\mathrm{d}}_{e^H}(X)=e^{i⟨\alpha ,H⟩}X}$

for all ${\displaystyle H\in \mathfrak{t}}$. Here ${\displaystyle ⟨\cdot ,\cdot ⟩}$ is a fixed inner product on ${\displaystyle \mathfrak{g}}$ that is invariant under the adjoint action of connected compact Lie groups.

The root system, as a subset of the Lie algebra ${\displaystyle \mathfrak{t}}$ of T, has all the usual properties of a root system, except that the roots may not span ${\displaystyle \mathfrak{t}}$. 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,

${\displaystyle W(T,G):=N_G(T)/C_G(T).}$

Fix a maximal torus ${\displaystyle T=T_0}$ 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 ${\displaystyle G=SU(n)}$ with ${\displaystyle T}$ being the diagonal subgroup of ${\displaystyle G}$. Then ${\displaystyle x\in G}$ belongs to ${\displaystyle N(T)}$ if and only if ${\displaystyle x}$ maps each standard basis element ${\displaystyle e_i}$ to a multiple of some other standard basis element ${\displaystyle e_j}$, that is, if and only if ${\displaystyle x}$ permutes the standard basis elements, up to multiplication by some constants. The Weyl group in this case is then the permutation group on ${\displaystyle n}$ 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:

${\displaystyle {\displaystyle {\int }_{G}f(g)dg=|W{|}^{-1}{\int }_T|\mathrm{\Delta }(t){|}^2{\int }_{G/T}f\left(yt{y}^{-1}\right)d[y]dt,}}$

where ${\displaystyle d[y]}$ is the normalized volume measure on the quotient manifold ${\displaystyle G/T}$ and ${\displaystyle dt}$ is the normalized Haar measure on T. Here Δ is given by the Weyl denominator formula and ${\displaystyle |W|}$ 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

${\displaystyle {\displaystyle {\int }_{G}f(g)dg=|W{|}^{-1}{\int }_{T}f(t)|\mathrm{\Delta }(t){|}^{2}dt.}}$

Consider as an example the case ${\displaystyle G=SU(2)}$, with ${\displaystyle T}$ being the diagonal subgroup. Then the Weyl integral formula for class functions takes the following explicit form:

${\displaystyle {\displaystyle {\int }_{SU(2)}f(g)dg=\frac{1}{2}{\int }_0^{2\pi }f\left(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(e^{i\theta },e^{-i\theta }\right)\right)4{\mathrm{s}\mathrm{i}\mathrm{n}}^2(\theta )\frac{d\theta }{2\pi }.}}$

Here ${\displaystyle |W|=2}$, the normalized Haar measure on ${\displaystyle T}$ is ${\displaystyle \frac{d\theta }{2\pi }}$, and ${\displaystyle \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(e^{i\theta },e^{-i\theta }\right)}$ denotes the diagonal matrix with diagonal entries ${\displaystyle e^{i\theta }}$ and ${\displaystyle e^{-i\theta }}$.