Lie Algebra Weight

Consider a collection of diagonal matrices H_1,...,H_k, which span a subspace h. Then the ith eigenvalue, i.e., the ith entry along the diagonal, is a linear functional on h, 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 h is Abelian and can be put into diagonal form. For example, consider the standard representation of the special linear Lie algebra sl_3(C) on C^3. Then

 H_1=[1  0 0; 0 -1 0; 0  0 0]


 H_2=[0  0 0; 0 -1 0; 0  0 1]

span the Cartan subalgebra h. There are three weights,




corresponding to the decomposition of

 C^3=<e_1> direct sum <e_2> direct sum <e_3>

into its eigenspaces. Note that alpha_1+alpha_2+alpha_3=0, because the matrices have zero matrix trace. The eigenvectors e_1,e_2,e_3 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 h^*. The set of all possible weights forms a weight lattice, which contains the root lattice. The Lie Algebra representations of g can be classified using the weight lattice.

See also

Cartan Matrix, Lie Algebra, Lie Algebra Root, Root System, Semisimple Lie Algebra, Weyl Chamber, Weyl Group

This entry contributed by Todd Rowland

Explore with Wolfram|Alpha

Cite this as:

Rowland, Todd. "Lie Algebra Weight." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications