# Weight of a representation of a Lie algebra

in a vector space $V$

A linear mapping $\alpha$ from the Lie algebra $L$ into its field of definition $k$ for which there exists a non-zero vector $x$ of $V$ such that for the representation $\rho$ one has

$$( \rho ( h) - \alpha ( h) 1) ^ {{n} _ {x,h}} ( x) = 0$$

for all $h \in L$ and some integer $n _ {x,h} > 0$ ( which in general depends on $x$ and $h$). Here 1 denotes the identity transformation of $V$. One also says in such a case that $\alpha$ is a weight of the $L$-module $V$ defined by the representation $\rho$. The set of all vectors $x \in V$ which satisfy this condition, together with zero, forms a subspace $V _ \alpha$, which is known as the weight subspace of the weight $\alpha$ (or corresponding to $\alpha$). If $V = V _ \alpha$, then $V$ is said to be a weight space or weight module over $L$ of weight $\alpha$.

If $V$ is a finite-dimensional module over $L$ of weight $\alpha$, its contragredient module (cf. Contragredient representation) $V ^ {*}$ is a weight module of weight $- \alpha$; if $V$ and $W$ are weight modules over $L$ of weights $\alpha$ and $\beta$, respectively, then their tensor product $V \otimes W$ is a weight module of weight $\alpha + \beta$. If $L$ is a nilpotent Lie algebra, a weight subspace $V _ \alpha$ of weight $\alpha$ in $V$ is an $L$-submodule of the $L$- module $V$. If, in addition,

$$\mathop{\rm dim} _ {k} V < \infty$$

and $\rho ( L)$ is a splitting Lie algebra of linear transformations of the module $V$, then $V$ can be decomposed into a direct sum of a finite number of weight subspaces of different weights:

$$V = V _ \sigma \oplus V _ \delta \oplus \dots \oplus V _ \tau$$

(the weight decomposition of $V$ with respect to $L$). If $L$ is a nilpotent subalgebra of a finite-dimensional Lie algebra $M$, considered as an $L$- module with respect to the adjoint representation ${ \mathop{\rm ad} } _ {M}$ of $M$ ( cf. Adjoint representation of a Lie group), and ${ \mathop{\rm ad} } _ {M} L$ is a splitting Lie algebra of linear transformations of $M$, then the corresponding weight decomposition of $M$ with respect to $L$,

$$M = M _ \alpha \oplus M _ \beta \oplus \dots \oplus M _ \gamma$$

is called the Fitting decomposition of $M$ with respect to $L$, the weights $\alpha , \beta \dots \gamma$ are called the roots, while the spaces $M _ \alpha , M _ \beta \dots M _ \gamma$ are called the root subspaces of $M$ with respect to $L$. If, in addition, one specifies the representation $\rho$ of the algebra $M$ in a finite-dimensional vector space $V$ for which $\rho ( L)$ is a splitting Lie algebra of linear transformations of $V$, and

$$V = V _ \sigma \oplus V _ \delta \oplus \dots \oplus V _ \tau$$

is the corresponding weight decomposition of $V$ with respect to $L$, then $\rho ( M _ \alpha )( V _ \sigma ) \subseteq V _ {\alpha + \sigma }$ if $\alpha + \sigma$ is a weight of $V$ with respect to $L$, and $\rho ( M _ \alpha )( V _ \sigma ) = 0$ otherwise. In particular, if $\alpha + \beta$ is a root, then $[ M _ \alpha , M _ \beta ] \subseteq M _ {\alpha + \beta }$, and $[ M _ \alpha , M _ \beta ] = 0$ otherwise. If $k$ is a field of characteristic zero, the weights $\sigma , \delta \dots \tau$ and the roots $\alpha , \beta \dots \gamma$ are linear functions on $L$ which vanish on the commutator subalgebra of $L$.

#### References

 [1] N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) [2] D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)

A set (algebra, Lie algebra, etc.) $L$ of linear transformations of a vector space over a field $k$ is called split or splitting if the characteristic polynomial of each of the transformations has all its roots in $k$, i.e. if $k$ contains a splitting field (cf. Splitting field of a polynomial) of the characteristic polynomial of each $h \in L$.
A representation $\rho : L \rightarrow \mathop{\rm End} ( V)$ of Lie algebras is split if $\rho ( L)$ is a split Lie algebra of linear transformations.