Lie Algebra Weight

A Lie algebra weight is a linear functional arising from a simultaneous eigenspace in a Lie algebra representation. To illustrate, 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 finite-dimensional Lie algebra representation of a finite-dimensional complex semisimple Lie algebra , in which case the Cartan subalgebra is abelian and its representing matrices can be simultaneously diagonalized. For example, consider the standard Lie algebra 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 , the nonzero 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 finite-dimensional Lie algebra representations of can be classified using the weight lattice.

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