Weight of a representation of a Lie algebra

in a vector space

A linear mapping from the Lie algebra into its field of definition for which there exists a non-zero vector of such that for the representation one has

for all and some integer (which in general depends on and ). Here 1 denotes the identity transformation of . One also says in such a case that is a weight of the -module defined by the representation . The set of all vectors which satisfy this condition, together with zero, forms a subspace , which is known as the weight subspace of the weight (or corresponding to ). If , then is said to be a weight space or weight module over of weight .

If is a finite-dimensional module over of weight , its contragredient module (cf. Contragredient representation) is a weight module of weight ; if and are weight modules over of weights and , respectively, then their tensor product is a weight module of weight . If is a nilpotent Lie algebra, a weight subspace of weight in is an -submodule of the -module . If, in addition,

and is a splitting Lie algebra of linear transformations of the module , then can be decomposed into a direct sum of a finite number of weight subspaces of different weights:

(the weight decomposition of with respect to ). If is a nilpotent subalgebra of a finite-dimensional Lie algebra , considered as an -module with respect to the adjoint representation of (cf. Adjoint representation of a Lie group), and is a splitting Lie algebra of linear transformations of , then the corresponding weight decomposition of with respect to ,

is called the Fitting decomposition of with respect to , the weights are called the roots, while the spaces are called the root subspaces of with respect to . If, in addition, one specifies the representation of the algebra in a finite-dimensional vector space for which is a splitting Lie algebra of linear transformations of , and

is the corresponding weight decomposition of with respect to , then if is a weight of with respect to , and otherwise. In particular, if is a root, then , and otherwise. If is a field of characteristic zero, the weights and the roots are linear functions on which vanish on the commutator subalgebra of .

References

[1]	N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))
[2]	D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)

Comments

A set (algebra, Lie algebra, etc.) of linear transformations of a vector space over a field is called split or splitting if the characteristic polynomial of each of the transformations has all its roots in , i.e. if contains a splitting field (cf. Splitting field of a polynomial) of the characteristic polynomial of each .

A representation of Lie algebras is split if is a split Lie algebra of linear transformations.

References

[a1]	N. Bourbaki, "Groupes et algèbres de Lie" , Hermann (1975) pp. Chapts. VII-VIII