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Cartan Subalgebras, Roots, and Dynkin Diagrams

The payoff: diagonalize a semisimple algebra against a maximal toral Cartan subalgebra and the nonzero eigenvalues — the roots — assemble into a finite, highly symmetric configuration. Root systems satisfy rigid integrality and reflection axioms, reduce to a Cartan matrix and a Dynkin diagram, and that diagram classifies every complex simple Lie algebra: A–B–C–D plus five exceptions.

Cartan subalgebras and the root space decomposition

Inside a semisimple L over ℂ, choose a Cartan subalgebra H: a maximal subalgebra that is abelian and consists of *semisimple* (diagonalizable) elements — a maximal torus. Because the elements of H commute and are diagonalizable, the operators ad(h) for h ∈ H are *simultaneously* diagonalizable on L. Decompose L into their common eigenspaces. The eigenvalue on each piece is a linear functional α ∈ H*, and the nonzero ones are the roots.

This gives the root space decomposition L = H ⊕ (⊕_{α∈Φ} L_α), where L_α = {x : [h,x]=α(h)x for all h ∈ H} and Φ ⊂ H* is the finite set of roots. Each root space L_α is one-dimensional. The bracket respects the grading: [L_α, L_β] ⊆ L_{α+β}. So the entire algebra is reconstructed from H, the roots, and these additive shift rules — an astonishing compression of a possibly large algebra into a finite combinatorial picture.

sl(3,C):  dim 8.  H = diagonal traceless matrices, dim 2.
  h = diag(a, b, c) with a+b+c = 0.
Root spaces: each off-diagonal E_ij (i != j) is a root vector.
  [h, E_ij] = (h_i - h_j) E_ij,   so the root is  alpha_ij(h) = h_i - h_j.
Let L_i be the functional h -> h_i.  Roots:
  Phi = { L_i - L_j : i != j }  =  six roots in the 2-dim space H*.

Simple roots:  alpha = L_1 - L_2 ,  beta = L_2 - L_3.
The six roots:  +-alpha, +-beta, +-(alpha+beta).
Lengths/angles via the Killing form:
  |alpha| = |beta|,  angle(alpha,beta) = 120 degrees.
This is the root system A_2 : a regular hexagon.

Cartan integers (the matrix):
  <alpha,beta> = 2(alpha,beta)/(beta,beta) = -1,  <beta,alpha> = -1
Cartan matrix A_2 = [ 2, -1; -1, 2 ].
Dynkin diagram:  two nodes joined by a single edge:   o---o
sl(3,ℂ) has root system A₂: six roots in a hexagon, two simple roots at 120°, Cartan matrix [2,−1;−1,2], Dynkin diagram o—o.

Root systems: integrality and reflections

Abstract a root system Φ in a Euclidean space E (the inner product coming from the Killing form). The axioms are spare but ferocious: (1) Φ is finite, spans E, and 0 ∉ Φ; (2) the only multiples of α ∈ Φ that are roots are ±α; (3) the reflection s_α through the hyperplane ⟂ α permutes Φ; (4) the Cartan integer ⟨β,α⟩ = 2(β,α)/(α,α) is an integer for all α,β ∈ Φ. Axiom (4) is the secret weapon: a continuous geometric object forced onto an integer lattice.

Dynkin diagrams and the classification

Pick a base of simple roots Δ ⊆ Φ: every root is a nonnegative or nonpositive integer combination of them. Record their pairwise Cartan integers in the Cartan matrix, then draw the Dynkin diagram: one node per simple root, with α and β joined by ⟨α,β⟩⟨β,α⟩ ∈ {0,1,2,3} edges, and an arrow pointing from the longer root to the shorter when lengths differ. The whole rigidity of the root system collapses into this small decorated graph.

The grand theorem: connected Dynkin diagrams are classified completely, and each corresponds to exactly one complex simple Lie algebra. The list is four infinite families — A_n (sl(n+1)), B_n (so(2n+1)), C_n (sp(2n)), D_n (so(2n)) — and five exceptionals E_6, E_7, E_8, F_4, G_2. That is the entire universe of complex simple Lie algebras. A continuous-symmetry classification problem has a fully discrete, finite answer; this is one of the most beautiful results in all of algebra.