# 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 $.

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**How to Cite This Entry:**

Weight space.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Weight_space&oldid=49195