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.