# Weight of a representation of a Lie algebra

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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 .

How to Cite This Entry:
Weight of a representation of a Lie algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weight_of_a_representation_of_a_Lie_algebra&oldid=17874
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article