Root systems and Dynkin diagrams

Choose a positive-definite invariant inner product on the real root span. Reflection in the hyperplane perpendicular to α sends β to β−2(β,α)α/(α,α) and preserves the root set. Crystallographic integrality requires the coefficient 2(β,α)/(α,α) to be an integer for roots α and β. Reducedness says the only scalar multiples of a root that are roots are itself and its negative. These requirements exclude many otherwise plausible finite arrow configurations. The geometric inner product is not the original complex Killing form treated as a Hermitian norm.

A positive system determines simple roots, which form a basis of the real root span and express each positive root as a nonnegative integral combination. Simple roots have nonpositive mutual pairings. Root strings are compatible with the sl(2) structure: if β−pα through β+qα is a string for nonproportional roots, then p−q equals the coroot pairing of β with α. This integrality is a consequence of finite strings, not an arbitrary lattice decoration added after drawing the roots.

Diagonal Cartan entries are two. Off-diagonal entries are nonpositive integers, and the product aᵢⱼaⱼᵢ is 0,1,2, or 3 for distinct simple roots in a finite root system. These cases correspond to no edge, a single edge, a double edge, or a triple edge. Multiple edges carry an arrow toward the shorter root. The product determines the angle while the ratio determines the squared-length ratio. A double line without its short-root orientation loses information needed to distinguish the two standard coroot conventions.

For example, type B₂ can be represented by long simple root e₁−e₂ and short simple root e₂ in an orthonormal plane. Their inner product is −1, with squared lengths two and one. Under this lesson's convention the Cartan matrix has first row (2,−1) and second row (−2,2). Transposing it exchanges the root and coroot viewpoints. B₂ and C₂ give isomorphic complex Lie algebras, a low-rank coincidence that should not be extrapolated to all higher ranks.

Connected finite Dynkin diagrams classify simple complex Lie algebras up to isomorphism. One nonredundant list uses Aₙ for n≥1, Bₙ for n≥2, Cₙ for n≥3, Dₙ for n≥4, and the exceptional E₆,E₇,E₈,F₄,G₂. Semisimple algebras correspond to disjoint unions, or direct sums of simple factors. Small-rank identifications explain the range restrictions. Establishing that each diagram produces an algebra uses generators and Serre relations; the root-system geometry alone is not the full construction proof.

The diagram does not classify real forms or determine the global topology of a Lie group integrating the complex algebra. Simply connected and adjoint groups can share the same Lie algebra while admitting different representation descent conditions. Nor does every generalized Cartan matrix define a finite-dimensional semisimple algebra; dropping the finite positive-definite root-system condition leads into other theories. A diagram is powerful precisely because its scope is controlled: finite reduced crystallographic roots and finite-dimensional complex semisimple structure.