Root System - Root Systems and Lie Theory

Root Systems and Lie Theory

Irreducible root systems classify a number of related objects in Lie theory, notably the

  • simple Lie groups (see the list of simple Lie groups), including the
  • simple complex Lie groups;
  • their associated simple complex Lie algebras; and
  • simply connected complex Lie groups which are simple modulo centers.

In each case, the roots are non-zero weights of the adjoint representation.

In the case of a simply connected simple compact Lie group G with maximal torus T, the root lattice can naturally be identified with Hom(T, T) and the coroot lattice with Hom(T, T); see Adams (1983).

For connections between the exceptional root systems and their Lie groups and Lie algebras see E8, E7, E6, F4, and G2.

Exceptional Lie groups
  • G2
  • F4
  • E6
  • E7
  • E8

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