Lie Algebra Root

The roots of a semisimple Lie algebra are the Lie algebra weights occurring in its adjoint representation. The set of roots form the root system, and are completely determined by . It is possible to choose a set of Lie algebra positive roots, every root is either positive or 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. For example, in the special Lie algebra of two by two matrices with zero matrix trace, has a basis given by the matrices

(1)

The adjoint representation is given by the brackets

(2)

(3)

so there are two roots of given by and . The Lie algebraic rank of is one, and it has one positive root.