Lie Algebra Root

A Lie algebra root of a finite-dimensional complex semisimple Lie algebra is a nonzero Lie algebra weight occurring in its adjoint representation with respect to a chosen Cartan subalgebra. The set of roots forms a root system, whose isomorphism type is 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.

For a real semisimple Lie algebra, this root-system description is applied to its complexification.

The simple roots can be considered as a linearly independent finite subset of Euclidean space, and they generate the root lattice. For example, the special linear Lie algebra of two by two matrices with zero matrix trace has a vector 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.