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^[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=\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, one 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:^[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:

All maximal tori in G are conjugate.^[3]
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.^[4]
The maximal tori in G are exactly the Lie subgroups corresponding to the maximal abelian subalgebras of ${\mathfrak {g}}$ ^[5] (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 ${\mathfrak {t}}$ .^[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):=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.^[7]
The Weyl group is generated by reflections about the roots of the associated Lie algebra.^[8] 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.^[9] 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.^[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 {\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:^[11]

\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 $e^{-i\theta }$ .

References

^ Hall 2015 Theorem 11.2
^ Hall 2015 Lemma 11.12
^ Hall 2015 Theorem 11.9
^ Hall 2015 Theorem 11.36 and Exercise 11.5
^ Hall 2015 Proposition 11.7
^ Hall 2015 Section 11.7
^ Hall 2015 Theorem 11.36
^ Hall 2015 Theorem 11.36
^ Hall 2015 Theorem 11.39
^ Hall 2015 Theorem 11.30 and Proposition 12.24
^ Hall 2015 Example 11.33

Adams, J. F. (1969), Lectures on Lie Groups, University of Chicago Press, ISBN 0226005305
Bourbaki, N. (1982), Groupes et Algèbres de Lie (Chapitre 9), Éléments de Mathématique, Masson, ISBN 354034392X
Dieudonné, J. (1977), Compact Lie groups and semisimple Lie groups, Chapter XXI, Treatise on analysis, vol. 5, Academic Press, ISBN 012215505X
Duistermaat, J.J.; Kolk, A. (2000), Lie groups, Universitext, Springer, ISBN 3540152938
Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666
Helgason, Sigurdur (1978), Differential geometry, Lie groups, and symmetric spaces, Academic Press, ISBN 0821828487
Hochschild, G. (1965), The structure of Lie groups, Holden-Day

[1] Hall 2015 Theorem 11.2

[2] Hall 2015 Lemma 11.12

[3] Hall 2015 Theorem 11.9

[4] Hall 2015 Theorem 11.36 and Exercise 11.5

[5] Hall 2015 Proposition 11.7

[6] Hall 2015 Section 11.7

[7] Hall 2015 Theorem 11.36

[8] Hall 2015 Theorem 11.36

[9] Hall 2015 Theorem 11.39

[10] Hall 2015 Theorem 11.30 and Proposition 12.24

[11] Hall 2015 Example 11.33

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