Properties:
Rank: —
dim(g): —
|Φ| roots: —
|W| (Weyl): —
Exponents: —
Coxeter h: —
Killing's classification (1888):
Every complex simple Lie algebra is one of:
Aₙ, Bₙ, Cₙ, Dₙ (n≥2), E₆, E₇, E₈, F₄, G₂
Cartan matrix:
Aᵢⱼ = 2⟨αᵢ,αⱼ⟩/⟨αⱼ,αⱼ⟩
Off-diagonal: 0,−1,−2,−3
Simply laced (all bonds single):
Aₙ, Dₙ, E₆, E₇, E₈
E₈ is special: 240 roots, dim=248, appears in string theory compactification
Every complex simple Lie algebra is one of:
Aₙ, Bₙ, Cₙ, Dₙ (n≥2), E₆, E₇, E₈, F₄, G₂
Cartan matrix:
Aᵢⱼ = 2⟨αᵢ,αⱼ⟩/⟨αⱼ,αⱼ⟩
Off-diagonal: 0,−1,−2,−3
Simply laced (all bonds single):
Aₙ, Dₙ, E₆, E₇, E₈
E₈ is special: 240 roots, dim=248, appears in string theory compactification