Cartan classification of simple Lie algebras
Statement
Élie Cartan completed the structure theory of finite-dimensional complex simple Lie algebras: every such algebra is isomorphic to one of the classical types (special linear, orthogonal, symplectic) or one of the five exceptional Lie algebras G₂, F₄, E₆, E₇, E₈, classified via root systems and Dynkin diagrams (later graphical form due to Dynkin, 1946). Cartan’s thesis and subsequent papers introduced Cartan subalgebras, the Killing form, and the root-space decomposition. This classification is foundational for Lie groups, differential geometry (symmetric spaces, moving frames), and particle physics (gauge groups). Proofs are pure algebra and formalizable in ZFC.
Formula
𝔤 simple / ℂ ⇒ type Aₙ…G₂
Equation
Root system ⇒ Cartan matrix ⇒ Dynkin diagram
Proof sketch
Cartan’s 1894 thesis Sur la structure des groupes de transformations finis et continus; later expositions in papers on simple groups and infinite-dimensional groups. Killing had begun the program; Cartan gave the complete rigorous classification.
Source
Cartan, É. Sur la structure des groupes de transformations finis et continus. Thèse, Paris, 1894. See also Cartan, Oeuvres complètes.
Domains
mathematics
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- Theorem: Cartan classification of simple Lie algebras
- Proof: Cartan classification of simple Lie algebras