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.
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