Root system
Introduction
A root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. Root systems are prevalent in the study of Lie groups, Lie algebras, and algebraic groups. They provide a crucial tool for the classification and representation theory of these algebraic structures.
Definition
A root system in a finite-dimensional Euclidean space E is a finite set Φ of non-zero vectors (called roots) that satisfies the following conditions:
1. The roots span E. 2. For each root α in Φ, the set Φ is closed under reflection through the hyperplane perpendicular to α. 3. The only scalar multiples of α in Φ are α itself and −α.
Properties
A root system Φ determines a vector space decomposition E = P ⊕ Q, where P is the span of Φ, and Q is the orthogonal complement of P. This decomposition is invariant under all reflections in Φ, and the reflection group W(Φ) generated by these reflections is a finite Coxeter group.
Classification
Root systems are classified up to isomorphism by their Dynkin diagrams. The classification theorem states that there are four infinite families of root systems (An, Bn, Cn, Dn), and five exceptional ones (E6, E7, E8, F4, G2).
Applications
Root systems are used in the classification and representation theory of semisimple Lie algebras, compact Lie groups, and algebraic groups. They also appear in the theory of integrable systems.