Root Systems of Semisimple Lie Algebras

Core Idea

A complex semisimple Lie algebra g decomposes under the adjoint action of a Cartan subalgebra h into h itself plus one-dimensional root spaces g_α, one for each root α in a finite set Φ ⊂ h*. The roots, with the inner product coming from the Killing form, form a root system: a finite spanning set of vectors that is closed under the reflections it defines and whose pairings 2(β, α)/(α, α) are integers. Root systems are rigid enough to classify completely by Dynkin diagrams, giving the four infinite families A_n, B_n, C_n, D_n and the five exceptional types E₆, E₇, E₈, F₄, G₂. This classification of root systems yields the classification of complex simple Lie algebras and organizes their representation theory.

How It's Best Learned

Work out sl₃(ℂ) in full: take h to be the traceless diagonal matrices, find the six roots eᵢ - eⱼ, and draw them as a regular hexagon in the plane. Use copies of sl₂ inside g, one for each pair ±α, to see why the pairings must be integers. Then study the axioms abstractly, list the possible angles between two roots, choose simple roots and build Dynkin diagrams for the rank-two systems A₂, B₂ and G₂ before stating the general classification.

Common Misconceptions

• A root is an element of the Lie algebra. (A root is a linear functional on h, an element of h*; the root vectors that span each g_α are the elements of g.)
• The angle between two roots can be anything. (The integrality condition forces the angle between two non-proportional roots to be 90°, 60°, 120°, 45°, 135°, 30° or 150°.)
• The root system depends on which Cartan subalgebra is chosen. (All Cartan subalgebras of a complex semisimple Lie algebra are conjugate, so the root system is determined up to isomorphism.)

Explainer

The representation theory of sl₂(ℂ) rests on one element h whose eigenvalues, the weights, organize everything. For a general complex semisimple Lie algebra g, the role of h is played by a Cartan subalgebra: a maximal abelian subalgebra consisting of elements that act diagonalizably in the adjoint representation. Since the elements of h commute, they can be diagonalized simultaneously on g. For each linear functional α on h, set g_α = {x ∈ g : [h, x] = α(h)x for all h in h}. The nonzero α with g_α ≠ 0 are the roots, and

g = h ⊕ (direct sum over α ∈ Φ of g_α).

Each root space is one-dimensional, and if α is a root, so is -α.

Take g = sl₃(ℂ), with h the traceless diagonal matrices diag(a₁, a₂, a₃). The matrix unit E_ij with i ≠ j satisfies [h, E_ij] = (aᵢ - aⱼ)E_ij. So the roots are the six functionals eᵢ - eⱼ, and dim g = 2 + 6 = 8. Drawn in the plane a₁ + a₂ + a₃ = 0, they form the vertices of a regular hexagon.

The Killing form gives an inner product on the real span of the roots, and the roots then satisfy four properties. They are finite, span the space, and exclude 0. The only multiples of α in Φ are ±α. The reflection s_α(β) = β - 2(β, α)/(α, α) α maps Φ to itself. And the numbers 2(β, α)/(α, α) are integers. A set with these properties is a root system. The integrality comes from sl₂: each pair ±α spans, together with a coroot in h, a copy of sl₂ inside g, and g becomes a finite-dimensional sl₂-representation whose weights must be integers.

These axioms are very restrictive. Multiplying the two integers 2(β, α)/(α, α) and 2(α, β)/(β, β) gives 4cos²θ, so the angle θ between roots can take only a few values, and length ratios are limited too. Choosing a set of simple roots, from which every root is a sum with all nonnegative or all nonpositive integer coefficients, and recording their angles in a Dynkin diagram, one finds that the irreducible root systems are exactly A_n, B_n, C_n, D_n, E₆, E₇, E₈, F₄ and G₂. The reflections generate a finite group, the Weyl group.

Root systems then organize representations. Weights of any finite-dimensional representation lie in a lattice determined by the roots, and the theorem of the highest weight says that irreducible representations correspond to dominant integral weights.