Page "Weight (representation theory)" Paragraph 8

If A is a Lie algebra, then the commutativity of the field and the anticommutativity of the Lie bracket imply that this map vanish on commutators: χ ()= 0.

A weight on a Lie algebra g over a field F is a linear map λ: g → F with λ ( y )= 0 for all x, y in g. Any weight on a Lie algebra g vanishes on the derived algebra and hence descends to a weight on the abelian Lie algebra g /.

Thus weights are primarily of interest for abelian Lie algebras, where they reduce to the simple notion of a generalized eigenvalue for space of commuting linear transformations.