advanced34 min studyLesson 499 in Mathematics

Compact Lie groups and maximal tori

Reduce compact connected symmetry to conjugate torus subgroups while distinguishing rank, dimension, and singular centralizers.

Before you begin

On this page

The question for today

What you’ll understand

State maximal-torus conjugacy and element coverage for compact connected groups.
Compute a torus and rank in unitary matrix groups.
Identify why connectedness and compactness cannot be dropped from the stated conclusions.

A large compact connected Lie group contains simpler commuting subgroups shaped like products of circles. A maximal torus is such a subgroup that cannot be enlarged to a larger torus inside the group. The remarkable theorem is that all maximal tori are conjugate and every group element belongs to one of them. This does not make the entire group abelian: different elements may require different conjugating matrices. It gives a common structural model for individual elements, with a rank that is often much smaller than the group's total dimension.

A torus is a closed connected compact abelian subgroup

A torus subgroup is isomorphic to a finite product of circles, including the zero-dimensional trivial torus. In a compact connected Lie group, maximal tori exist, every torus is contained in one, and any two maximal tori are conjugate. Therefore their dimensions agree; that common dimension is the rank. Maximal means maximal among torus subgroups, not maximal among all proper subgroups and not necessarily the subgroup with the largest matrix size.

Every element of a compact connected Lie group lies in some maximal torus. Fixing one maximal torus T, every element is consequently conjugate to a member of T. Since the torus exponential is onto, this also implies that the exponential of a compact connected group is onto. Its failure of global injectivity remains: torus periods persist. A conjugacy statement is not a unique coordinate parametrization, and repeated eigenvalues can make the conjugating choices particularly nonunique.

G\ \text{compact and connected},\quad T\ \text{maximal torus}\quad\Longrightarrow\quad G=\bigcup_{g\in G}gTg^{-1},\qquad \operatorname{rank}G=\dim T

All maximal tori are conjugate under these hypotheses. The union is a structural covering by conjugate subgroups, not a disjoint union or a claim that G itself is a torus.↔ Scroll wide equations horizontally.

Unitary diagonalization gives a concrete model

In U-n, diagonal unitary matrices form a maximal torus with n independent phases, so the rank is n while the real group dimension is n squared. The spectral theorem diagonalizes every unitary matrix by a unitary change of basis, providing the element-conjugacy statement in this family. In SU-n, determinant one imposes one phase relation and the rank becomes n minus one. These phase relations concern a torus and do not reduce the entire group dimension by the same large amount.

The centralizer of a torus in a compact connected group is the torus itself when the torus is maximal. The centralizer of an individual element can be larger. In SU-two a diagonal element with two distinct eigenvalues has the diagonal torus as its centralizer, while the central elements plus and minus identity commute with the entire group. This variation is related to regular versus singular elements and is one source of nonuniqueness when using conjugation to choose torus coordinates.

T_{U(n)}=\{\operatorname{diag}(e^{i\theta_1},\ldots,e^{i\theta_n})\},\qquad T_{SU(n)}=\{t\in T_{U(n)}:\theta_1+\cdots+\theta_n\equiv0\pmod{2\pi}\}

The ranks are n and n−1 respectively. These torus dimensions differ from the real group dimensions n² and n²−1.↔ Scroll wide equations horizontally.

Keep both compactness and connectedness visible

The additive real line is a connected noncompact Lie group and has no nontrivial compact torus subgroup, so torus coverage cannot be asserted for all connected groups. Conversely, the reflection component of O-two is outside every connected torus because each torus lies in the identity component SO-two. Thus compactness alone does not give the stated every-element conclusion for a disconnected group. More general noncompact and disconnected structure theorems require additional language.

Averaging over a compact group supplies an invariant inner product on its Lie algebra, which supports geometric proofs of maximal-torus facts. We use the theorem here rather than pretend that unitary diagonalization alone proves it for every compact group. In applications, first identify the actual group, its component structure, and its torus periods; only then use rank or phase coordinates. The next lesson explains the remaining finite ambiguity in those coordinates through roots and the Weyl group.

Work it through

Diagonalize a unitary element without confusing rank and dimension

For M=[[0,1],[−1,0]] in U(2), find an orthonormal eigenbasis and its torus representative. State the rank and real dimension of U(2).

The normalized vectors $v_+=(1,i)/\sqrt2$ and $v_-=(1,-i)/\sqrt2$ have unit norm and zero Hermitian inner product. Multiplication gives Mv₊=iv₊ and Mv₋=−iv₋.
With U having those vectors as columns, U is unitary and $U^*MU=\operatorname{diag}(i,-i)$ . The diagonal lies in the standard two-dimensional torus. Its determinant is one, so this particular element also belongs to SU(2).
The group U(2) has real dimension four and rank two. SU(2) instead has dimension three and rank one. The fact that one selected element lies in a one-dimensional SU(2) torus does not change the rank or dimension of the larger ambient U(2).

The result

The torus organizes eigenphases of an element, while rank and total dimension remain distinct invariants of the specified group.

The living notebook

Explore the model

Interactive lab

Interactive lab · quantum

Inspect a diagonal SU(2) phase action on one state

Pause, Reset, and set mixing angle θ=90° and relative phase φ=0°. Repeat from Reset with both θ=90° and φ=90°. Predict unchanged basis probabilities one half and a rotated equatorial Bloch direction. For the state (|0⟩+|1⟩)/√2, diag(eⁱᵅ,e⁻ⁱᵅ) changes relative phase by −2α, so α=−45° gives the second setting. This normalized-state instrument is not a maximal-torus classifier or a complete SU(2) action simulator.

Bloch vector / P(0)P(1)Phase latitude

Probability |0⟩: 71.1%
Probability |1⟩: 28.9%
Bloch (x, y, z): 0.78, 0.45, 0.42

Mixing angle θ65°Relative phase φ30°

t = 0 s

A normalized two-level pure state, not an interacting field simulation. Playback adds one radian per second to the relative phase; basis probabilities stay fixed. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA torus phase can change relative geometry while preserving selected probabilities; that observation does not imply the full group is abelian.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Conjugation supplies coverage by maximal tori. It does not make every element lie in the same subgroup, identify rank with dimension, or remove torus periods from the exponential map.

Your turn

Try it yourself

In SU(2), consider $t=\operatorname{diag}(e^{i\theta},e^{-i\theta})$ . Describe its centralizer when θ is neither 0 nor π modulo 2π, and compare θ=0 and θ=π.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Distinct eigenvalues force a commuting matrix to preserve both coordinate eigenspaces. At the exceptional angles the matrix is scalar.

Compare with the solution

For the stated nonexceptional angles the two eigenvalues are distinct. A commuting unitary matrix is therefore diagonal, and determinant one leaves exactly the standard circle torus. At θ=0, t=I, and at θ=π, t=−I. Both are central, so their centralizer is all of SU(2). The torus has dimension one, while the exceptional centralizers have dimension three.

Keep the idea

Take it with you

Compact connected groups have conjugate maximal tori containing every element up to conjugation.
Rank counts torus dimensions rather than all group degrees of freedom.
Disconnected components and noncompact groups require conclusions beyond this theorem.

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