Simple root

Fix a root system $\Phi\subset V$ (see root system ) together with a choice of positive roots $\Phi^+\subset\Phi$.

A root $\alpha\in\Phi^+$ is called simple if it cannot be written as a sum of two positive roots:

The set of simple roots is usually denoted $\Delta\subset\Phi^+$.

Key structural facts (standard in root system theory):

• $\Delta$ is a basis of $V$ (in particular, the simple roots are linearly independent).
• Every positive root is a nonnegative integer combination of simple roots: $\Phi^+ \subset \Big\{\sum_{\alpha\in\Delta} n_\alpha\,\alpha \;:\; n_\alpha\in\mathbb Z_{\ge 0}\Big\}.$

In semisimple Lie theory (see roots of a Lie algebra and root space decomposition ), choosing $\Delta$ is the combinatorial input for building the Cartan matrix and Dynkin diagram . In representation theory, simple roots control highest weights (see highest weight and highest weight theorem ).