Weight of a representation of a Lie algebra

in a vector space $ V $

A linear mapping $ \alpha $ from the Lie algebra $ L $ into its field of definition $ k $ for which there exists a non-zero vector $ x $ of $ V $ such that for the representation $ \rho $ one has

$$ ( \rho ( h) - \alpha ( h) 1) ^ {{n} _ {x,h}} ( x) = 0 $$

for all $ h \in L $ and some integer $ n _ {x,h} > 0 $ ( which in general depends on $ x $ and $ h $). Here 1 denotes the identity transformation of $ V $. One also says in such a case that $ \alpha $ is a weight of the $ L $-module $ V $ defined by the representation $ \rho $. The set of all vectors $ x \in V $ which satisfy this condition, together with zero, forms a subspace $ V _ \alpha $, which is known as the weight subspace of the weight $ \alpha $ (or corresponding to $ \alpha $). If $ V = V _ \alpha $, then $ V $ is said to be a weight space or weight module over $ L $ of weight $ \alpha $.

If $ V $ is a finite-dimensional module over $ L $ of weight $ \alpha $, its contragredient module (cf. Contragredient representation) $ V ^ {*} $ is a weight module of weight $ - \alpha $; if $ V $ and $ W $ are weight modules over $ L $ of weights $ \alpha $ and $ \beta $, respectively, then their tensor product $ V \otimes W $ is a weight module of weight $ \alpha + \beta $. If $ L $ is a nilpotent Lie algebra, a weight subspace $ V _ \alpha $ of weight $ \alpha $ in $ V $ is an $ L $-submodule of the $ L $- module $ V $. If, in addition,

$$ \mathop{\rm dim} _ {k} V < \infty $$

and $ \rho ( L) $ is a splitting Lie algebra of linear transformations of the module $ V $, then $ V $ can be decomposed into a direct sum of a finite number of weight subspaces of different weights:

$$ V = V _ \sigma \oplus V _ \delta \oplus \dots \oplus V _ \tau $$

(the weight decomposition of $ V $ with respect to $ L $). If $ L $ is a nilpotent subalgebra of a finite-dimensional Lie algebra $ M $, considered as an $ L $- module with respect to the adjoint representation $ { \mathop{\rm ad} } _ {M} $ of $ M $ ( cf. Adjoint representation of a Lie group), and $ { \mathop{\rm ad} } _ {M} L $ is a splitting Lie algebra of linear transformations of $ M $, then the corresponding weight decomposition of $ M $ with respect to $ L $,

$$ M = M _ \alpha \oplus M _ \beta \oplus \dots \oplus M _ \gamma $$

is called the Fitting decomposition of $ M $ with respect to $ L $, the weights $ \alpha , \beta \dots \gamma $ are called the roots, while the spaces $ M _ \alpha , M _ \beta \dots M _ \gamma $ are called the root subspaces of $ M $ with respect to $ L $. If, in addition, one specifies the representation $ \rho $ of the algebra $ M $ in a finite-dimensional vector space $ V $ for which $ \rho ( L) $ is a splitting Lie algebra of linear transformations of $ V $, and

$$ V = V _ \sigma \oplus V _ \delta \oplus \dots \oplus V _ \tau $$

is the corresponding weight decomposition of $ V $ with respect to $ L $, then $ \rho ( M _ \alpha )( V _ \sigma ) \subseteq V _ {\alpha + \sigma } $ if $ \alpha + \sigma $ is a weight of $ V $ with respect to $ L $, and $ \rho ( M _ \alpha )( V _ \sigma ) = 0 $ otherwise. In particular, if $ \alpha + \beta $ is a root, then $ [ M _ \alpha , M _ \beta ] \subseteq M _ {\alpha + \beta } $, and $ [ M _ \alpha , M _ \beta ] = 0 $ otherwise. If $ k $ is a field of characteristic zero, the weights $ \sigma , \delta \dots \tau $ and the roots $ \alpha , \beta \dots \gamma $ are linear functions on $ L $ which vanish on the commutator subalgebra of $ L $.

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.) $ L $ of linear transformations of a vector space over a field $ k $ is called split or splitting if the characteristic polynomial of each of the transformations has all its roots in $ k $, i.e. if $ k $ contains a splitting field (cf. Splitting field of a polynomial) of the characteristic polynomial of each $ h \in L $.

A representation $ \rho : L \rightarrow \mathop{\rm End} ( V) $ of Lie algebras is split if $ \rho ( L) $ is a split Lie algebra of linear transformations.

References

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