Roots, Weyl groups, and compact group structure
Extract finite reflection data from a compact connected group while retaining the lattice and central quotient that distinguish global groups.
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The question for today
What you’ll understand
- Define roots through the complexified adjoint action of a maximal torus.
- Compute a Weyl action using a normalizer representative.
- Explain why roots alone do not determine the global compact group without lattice and central information.
A maximal torus turns continuous symmetry into commuting phases, but conjugation leaves a finite ambiguity in how those phases are chosen. Roots and the Weyl group organize that ambiguity. Roots are nontrivial torus characters appearing in the complexified adjoint representation; the Weyl group acts by the corresponding reflections. For compact connected groups, these data describe the semisimple directions with striking efficiency. The center and the torus's character lattice still matter: groups with the same tangent algebra can have different global topology and different representations that descend.
Diagonalize the torus action on the algebra
Let G be compact connected with maximal torus T and real Lie algebra g. Complexify the algebra and decompose it into weight spaces for the adjoint action of T. The zero-weight part is the complexified torus algebra. The nonzero weights are characters of T called roots, and their root spaces are one-dimensional complex spaces in this compact reductive setting. Differentiating a character gives a purely imaginary functional on the real torus algebra; dividing by i identifies the associated real root functional.
Roots vanish on the central torus directions, so they need not span the full dual torus algebra if G has a positive-dimensional center. On the semisimple part they form a reduced crystallographic root system. Distinguish a character, its differential, and its coordinates in a chosen basis: factors such as two in SU-two reflect the actual torus parametrization and cannot be removed without also changing the associated lattice. The complex root space is not itself a real one-dimensional tangent direction.
A normalizer quotient gives a finite reflection group
The normalizer of T consists of elements that conjugate T to itself. For a compact connected group, the centralizer of T equals T, and the Weyl group is the finite quotient N-G(T)/T. It acts faithfully on the torus and preserves the root system. On the real span of roots, after choosing an invariant inner product, it is generated by root reflections. On central directions its reflection action is trivial. These statements rely on the compact connected setting fixed at the beginning.
Conjugate elements of T are related by a Weyl-group element, so the conjugacy classes of G are described by T modulo this finite action. That quotient is not generally a smooth manifold everywhere: fixed points of Weyl elements produce singular orbit behavior. The reflection formula concerns a linear torus-algebra space, whereas the torus itself also has a period lattice. Keeping both the finite reflection and the periodic identifications is necessary when drawing a global parameter region.
Global groups retain information beyond a root picture
For a compact connected semisimple group, the root lattice Q sits inside the character lattice X-star(T), which sits inside the weight lattice P after the usual common Lie-algebra identification. The simply connected form has character lattice P and the adjoint form has Q. Intermediate lattices encode possible finite central quotients. SU-two and SO-three share the same local semisimple algebra and A-one root system but differ by the central kernel plus and minus identity. A root diagram alone cannot distinguish them.
More generally, every compact connected Lie group is a quotient of a product of a torus and a compact simply connected semisimple group by a finite central subgroup. The torus dimension records central continuous directions, while the finite quotient records additional global identifications. Root data and lattice information organize this structure; we are stating that theorem, not proving a classification from the single examples below. Nonconnected groups bring further component actions beyond the compact connected framework of this lesson.
Work it through
Find the SU(2) roots and its two-element Weyl action
For in SU(2), compute its adjoint weights on E₁₂ and E₂₁. Use n=[[0,−1],[1,0]] to identify the Weyl action.
Conjugation multiplies E₁₂ by and E₂₁ by . Thus the roots are the characters with integer weights plus two and minus two relative to the fundamental torus character . The diagonal traceless direction has zero weight.
The unitary determinant-one matrix n conjugates diag(z,z⁻¹) to diag(z⁻¹,z), so its action sends θ to −θ. It normalizes T but does not lie in T. Also n²=−I belongs to T, making its nontrivial quotient class have order two.
Every conjugation preserving this one-dimensional torus can only preserve or interchange its two coordinate eigenspaces, giving exactly these two Weyl classes. In the common SU(2) torus coordinate, characters descending to SO(3)=SU(2)/{±I} have even integer weights. The same roots ±2 coexist with a different allowable character lattice.
The reflection group has two elements, while the global distinction between SU(2) and SO(3) remains in the character lattice and central quotient.
The living notebook
Explore the model
Interactive lab · vectors
Inspect a reflection with a fixed hyperplane
Pause, Reset, then set angle=30° and magnitude=2, giving vector (√3,1). Repeat from Reset with both angle=−30° and magnitude=2, giving (√3,−1). Predict preserved length and unchanged horizontal component. This is the Euclidean reflection with vertical normal, illustrating the root-reflection formula in a plane; it is not a computed SU(3) root system or a classifier of compact groups.
- x component
- 1.393
- y component
- 0.975
- Magnitude
- 1.7
Playback traces a fixed vector and its projections. Angle and magnitude set the geometry; the moving marker does not change them. 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 Weyl reflection preserves an invariant inner product while changing a root-normal component; global torus periods remain additional data.
Pause & reason
Check your understanding
Your turn
Try it yourself
For SU(3), use the diagonal torus with infinitesimal angles θ₁+θ₂+θ₃=0. State its rank, roots, and Weyl-group order. Reflect v=(2,−1,−1) in the root α=e₁−e₂ using the standard Euclidean inner product.
Take a moment to make a prediction or work it out on paper. Open a hint when you need one.
Give me a hint
Conjugation of Eᵢⱼ has weight θᵢ−θⱼ. The corresponding reflection exchanges coordinates i and j.
Compare with the solution
The rank is two. The six roots are eᵢ−eⱼ for distinct indices i,j in {1,2,3}, restricted to the sum-zero plane. The Weyl group is S₃, of order six. Since (v,α)=3 and (α,α)=2, the reflection gives v−3α=(−1,2,−1), exchanging the first two entries and preserving the sum-zero condition and squared norm six.
Keep the idea
Take it with you
- Roots are nontrivial adjoint torus characters, with central directions excluded from their span.
- The finite normalizer quotient acts by root reflections in the compact connected setting.
- Character lattices and finite central quotients retain global information absent from roots alone.
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