Weight space

A finite-dimensional space $ V $ carrying a representation $ \rho $ of a Lie algebra $ L $ over a field $ F $ and satisfying the following condition: there exists a function $ \alpha : L \rightarrow F $ such that for any $ x \in V $, $ l \in L $,

$$ ( l ^ \rho - \alpha ( l) 1) ^ {k} x = 0 $$

for some positive integer $ k $. The function $ \alpha $ is called the weight. The tensor product $ \rho _ {1} \otimes \rho _ {2} $ of two representations $ \rho _ {1} , \rho _ {2} $ of $ L $ with weight spaces $ V _ {1} , V _ {2} $ which have weights $ \alpha _ {1} $ and $ \alpha _ {2} $, respectively, is the representation of $ L $ in the space $ V _ {1} \otimes V _ {2} $, which is also a weight space and has the weight $ \alpha _ {1} + \alpha _ {2} $. On passing from the representation $ \rho $ to the contragredient representation $ \rho ^ {*} $, the space $ V $ is replaced by the adjoint space $ V ^ {*} $, and $ \alpha $ is replaced by $ - \alpha $.

Comments

See also Weight of a representation of a Lie algebra.