Root System - Dual Root System and Coroots

Dual Root System and Coroots

See also: Langlands dual group

If Φ is a root system in V, the coroot αV of a root α is defined by

The set of coroots also forms a root system ΦV in V, called the dual root system (or sometimes inverse root system). By definition, αV V = α, so that Φ is the dual root system of ΦV. The lattice in V spanned by ΦV is called the coroot lattice. Both Φ and ΦV have the same Weyl group W and, for s in W,

If Δ is a set of simple roots for Φ, then ΔV is a set of simple roots for ΦV.

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