The roots of a semisimple Lie algebra g are the Lie algebra weights occurring in its adjoint representation. The set of roots form the root system, and are completely determined by g. It is possible to choose a set of Lie algebra positive roots, every root α is either positive or -α is positive. The Lie algebra simple roots are the positive roots which cannot be written as a sum of positive roots. The simple roots can be considered as a linearly independent finite subset of Euclidean space, and they generate the root lattice.

Cartan matrix | Lie algebra | Lie algebra weight | semisimple Lie algebra | Weyl group