Positive root

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

1. $\Phi$ is the disjoint union $\Phi = \Phi^+ \sqcup (-\Phi^+)$, and
2. if $\alpha,\beta \in \Phi^+$ and $\alpha+\beta \in \Phi$, then $\alpha+\beta \in \Phi^+$.

Equivalently, choose a linear functional $\lambda \in V^*$ such that $\lambda(\alpha)\neq 0$ for all $\alpha\in\Phi$, and set

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

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