Knowlpedia

Simple root

A minimal positive root; simple roots form a basis for the root system and generate all positive roots.

Fix a root system Φ⊂V\Phi\subset V (see ) together with a choice of Φ+⊂Φ\Phi^+\subset\Phi.

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

α≠β+γfor all β,γ∈Φ+.\alpha \neq \beta+\gamma \quad\text{for all }\beta,\gamma\in\Phi^+.

The set of simple roots is usually denoted Δ⊂Φ+\Delta\subset\Phi^+.

Key structural facts (standard in root system theory):

  • Δ\Delta is a basis of VV (in particular, the simple roots are linearly independent).
  • Every positive root is a nonnegative integer combination of simple roots:
    Φ+⊂{∑α∈Δnα α  :  nα∈Z≥0}.\Phi^+ \subset \Big\{\sum_{\alpha\in\Delta} n_\alpha\,\alpha \;:\; n_\alpha\in\mathbb Z_{\ge 0}\Big\}.

In semisimple Lie theory (see and ), choosing Δ\Delta is the combinatorial input for building the and . In representation theory, simple roots control highest weights (see and ).