Maximal torus

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

${\displaystyle T=\left\{\operatorname {diag} \left(e^{i\theta _{1}},e^{i\theta _{2}},\dots ,e^{i\theta _{n}}\right):\forall j,\theta _{j}\in \mathbb {R} \right\}.}$

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 ${\displaystyle 2\times 2}$ 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.

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:[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 ${\displaystyle {\mathfrak {g}}}$[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.

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 ${\displaystyle \alpha \neq 0}$ and there exists a nonzero ${\displaystyle X\in {\mathfrak {g}}_{\mathbb {C} }}$ such that

${\displaystyle \mathrm {Ad} _{e^{H}}(X)=e^{i\langle \alpha ,H\rangle }X}$

for all ${\displaystyle H\in {\mathfrak {t}}}$. Here ${\displaystyle \langle \cdot ,\cdot \rangle }$ 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 ${\displaystyle {\mathfrak {t}}}$.[6] The root system is a key tool in understanding the classification and representation theory of G.

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.[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 ${\displaystyle G=SU(n)}$ 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.

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:

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

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.[10] 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

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

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:[11]

${\displaystyle \displaystyle {\int _{SU(2)}f(g)\,dg={\frac {1}{2}}\int _{0}^{2\pi }f\left(\mathrm {diag} \left(e^{i\theta },e^{-i\theta }\right)\right)\,4\,\mathrm {sin} ^{2}(\theta )\,{\frac {d\theta }{2\pi }}.}}$

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

• Compact group
• Cartan subgroup
• Cartan subalgebra
• Toral Lie algebra
• Bruhat decomposition
• Weyl character formula
• Representation theory of a connected compact Lie group

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• 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, 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, 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