Classification of simple Lie algebras through crystallographic patterns
A few arrows in a perfectly symmetric star hold the whole structure of a Lie algebra. Reflect them, grow them from a diagram, and project the 240 roots of E₈.
Click any root to reflect the whole star across the mirror perpendicular to it. Notice that every reflection maps the six roots onto themselves.
Root systems are finite sets of vectors in a Euclidean space that encode the structure of semisimple Lie algebras. They arise from the eigenvalues of the adjoint representation — the roots are the “weights” by which the algebra acts on itself. The remarkable fact is that these root systems are completely classified: four infinite families (A, B, C, D) and five exceptional cases (E₆, E₇, E₈, F₄, G₂).
The root system of A₂ (= sl(3)) consists of six roots forming a regular hexagonal star. Click any root to apply a Weyl reflection — a mirror-sweep animation shows how the root system maps to itself under these reflections. Positive roots glow rose, negative roots glow blue. The Weyl group of A₂ is S₃ (the symmetric group on 3 elements), with 6 elements.
Key insight: Root systems have crystallographic symmetry — every root reflection maps roots to roots. This rigid constraint is what makes the classification possible and forces the angles between roots to be 30°, 45°, 60°, 90°, 120°, 135°, or 150°.
Browse the four rank-2 systems, up to the 12-root G₂. Toggle Weyl chambers, or switch to builder mode, place simple roots, and let reflections fill in the rest.
Explore all rank-2 root systems: A₁×A₁ (4 roots), A₂ (6 roots), B₂ (8 roots), and G₂ (12 roots — the double hexagon showpiece). Toggle Weyl chambers to see the fundamental domain of the reflection group. In builder mode, place simple roots and generate the full system by applying reflections.
Add nodes and join them with single, double, or triple bonds. The builder checks your diagram against the classification as you go; use the presets to see every family.
Dynkin diagrams distill all this information into simple graphs. Each node represents a simple root, and edges encode the angles between them. The classification of Dynkin diagrams — and hence of simple Lie algebras — is one of the most beautiful results in all of mathematics.
Build Dynkin diagrams by adding nodes and connecting them with single, double, or triple bonds. The system validates your diagram in real time against the classification theorem — if it matches a valid type, you'll see a “classification complete” celebration. Use presets to explore all the classical families and exceptional diagrams.
Pick a diagram and press play: it becomes a Cartan matrix, then simple roots, and then the full root system grows outward in waves of reflections.
Watch the step-by-step process: a Dynkin diagram determines a Cartan matrix, which determines the simple roots, which generate the full root system via Weyl reflections. Roots “grow” outward in waves of color like crystal formation, revealing the beautiful geometry hidden in the combinatorial data of the diagram.
The 240 roots of E₈, flattened from 8D onto the Coxeter plane, show 30-fold symmetry. Turn the two angle sliders to tilt the projection and see other views.
The exceptional Lie algebra E₈ has 240 roots living in 8-dimensional space. By projecting onto a carefully chosen 2D plane (the Coxeter projection), the 30-fold symmetry of E₈ is revealed in a stunning pattern. Control the projection plane to explore different views of this extraordinary object. Rose/coral gradient layers show the depth structure.
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