Lie Algebra Weight

Consider a collection of diagonal matrices , which span a subspace . Then the th eigenvalue, i.e., the th entry along the diagonal, is a linear functional on , and is called a weight.

The general setting for weights occurs in a Lie algebra representation of a semisimple Lie algebra, in which case the Cartan subalgebra is Abelian and can be put into diagonal form. For example, consider the standard representation of the special linear Lie algebra on . Then

(1)

and

(2)

span the Cartan subalgebra . There are three weights,

(3)

(4)

and

(5)

corresponding to the decomposition of

(6)

into its eigenspaces. Note that , because the matrices have zero matrix trace. The eigenvectors are called weight vectors, and the corresponding eigenspaces are called weight spaces.

In the important special case of the adjoint representation of a semisimple Lie algebra, the weights are called Lie algebra roots and the weight space is called the root space. The roots generate a discrete lattice, called the root lattice, in the dual vector space . The set of all possible weights forms a weight lattice, which contains the root lattice. The Lie Algebra representations of can be classified using the weight lattice.