# 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 $- \alpha$.