View source for Weight space

<!--
w0975601.png
$#A+1 = 25 n = 0
$#C+1 = 25 : ~/encyclopedia/old_files/data/W097/W.0907560 Weight space
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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====
See also [[Weight of a representation of a Lie algebra|Weight of a representation of a Lie algebra]].