Maximal torus explained

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[1] 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=

i\theta1
\left\{\operatorname{diag}\left(e
i\theta2
,e
i\thetan
,...,e

\right):\forallj,\thetaj\inR\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, one maximal torus consists of all block-diagonal matrices with

2 x 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

akg

be the Lie algebra of G. The first main result is the torus theorem, which may be formulated as follows:[2]

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:

akg

[5] (cf. Cartan subalgebra)

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

akt

denote the Lie algebra of T, let

akg

denote the Lie algebra of

G

, and let

akgC:=akgiakg

denote the complexification of

akg

. Then we say that an element

\alpha\inakt

is a root for G relative to T if

\alpha0

and there exists a nonzero

X\inakgC

such that
Ad
eH

(X)=ei\langle\alpha,H\rangleX

for all

H\inakt

. Here

\langle,\rangle

is a fixed inner product on

akg

that is invariant under the adjoint action of connected compact Lie groups.

The root system, as a subset of the Lie algebra

akt

of T, has all the usual properties of a root system, except that the roots may not span

akt

.[6] 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):=NG(T)/CG(T).

Fix a maximal torus

T=T0

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.

We now list some consequences of these main results.

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\inG

belongs to

N(T)

if and only if

x

maps each standard basis element

ei

to a multiple of some other standard basis element

ej

, 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{\intGf(g)dg=|W|-1\intT

2\int
|\Delta(t)|
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.[10] 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{\intGf(g)dg=|W|-1\intTf(t)|\Delta(t)|2dt.}

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:[11]

\displaystyle{\intSU(2)f(g)dg=

1
2
2\pi
\int
0

f\left(diag\left(ei\theta,e-i\theta\right)\right)4sin2(\theta)

d\theta
2\pi

.}

Here

|W|=2

, the normalized Haar measure on

T

is
d\theta
2\pi
, and

diag\left(ei\theta,e-i\theta\right)

denotes the diagonal matrix with diagonal entries

ei\theta

and

e-i\theta

.

See also

Notes and References

  1. Theorem 11.2
  2. Lemma 11.12
  3. Theorem 11.9
  4. Theorem 11.36 and Exercise 11.5
  5. Proposition 11.7
  6. Section 11.7
  7. Theorem 11.36
  8. Theorem 11.36
  9. Theorem 11.39
  10. Theorem 11.30 and Proposition 12.24
  11. Example 11.33