Knowlpedia

Positive root

A choice of “half” of a root set, compatible with addition, used to organize roots into positive and negative.

Let Φ\Phi be a root system in a real inner product space VV (see ). A positive system (or set of positive roots) is a subset Φ+⊂Φ\Phi^+ \subset \Phi such that:

  1. Φ\Phi is the disjoint union Φ=Φ+⊔(−Φ+)\Phi = \Phi^+ \sqcup (-\Phi^+), and
  2. if α,β∈Φ+\alpha,\beta \in \Phi^+ and α+β∈Φ\alpha+\beta \in \Phi, then α+β∈Φ+\alpha+\beta \in \Phi^+.

In the Lie algebra setting, for a semisimple Lie algebra with a , the roots (see ) can be split into positive and negative ones, producing a triangular decomposition (see ). This choice is essential for defining , constructing highest-weight theory, and stating results such as the .

Equivalent characterizations

Equivalently, choose a linear functional λ∈V∗\lambda \in V^* such that λ(α)≠0\lambda(\alpha)\neq 0 for all α∈Φ\alpha\in\Phi, and set

Φ+  =  {α∈Φ:λ(α)>0}.\Phi^+ \;=\; \{\alpha\in\Phi : \lambda(\alpha)>0\}.

Geometrically, this corresponds to choosing a Weyl chamber for the hyperplane arrangement {α⊥}α∈Φ\{\alpha^\perp\}_{\alpha\in\Phi}; changing the choice is controlled by the action.