Difference between revisions of "Weight space"
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| − | + | A finite-dimensional space $ V $ | |
| + | carrying a representation $ \rho $ | ||
| + | of a [[Lie algebra|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==== | ====Comments==== | ||
See also [[Weight of a representation of a Lie algebra|Weight of a representation of a Lie algebra]]. | See also [[Weight of a representation of a Lie algebra|Weight of a representation of a Lie algebra]]. | ||
Latest revision as of 08:29, 6 June 2020
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
How to Cite This Entry:
Weight space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weight_space&oldid=49195
Weight space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weight_space&oldid=49195
This article was adapted from an original article by E.N. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article