Editing Weight (representation theory) (section)

===Action of the root vectors===
For the [[Adjoint representation of a Lie algebra|adjoint representation]] <math>\mathrm{ad} : \mathfrak{g}\to \operatorname{End}(\mathfrak{g})</math> of  <math>\mathfrak g</math>, the space over which the representation acts is the Lie algebra itself. Then the nonzero weights are called '''[[root system|roots]]''', the weight spaces are called '''root spaces''', and the weight vectors, which are thus elements of <math>\mathfrak{g}</math>, are called '''root vectors'''. Explicitly, a linear functional <math>\alpha</math> on the Cartan subalgebra <math>\mathfrak h</math> is called a root if <math>\alpha\neq 0</math> and there exists a nonzero <math>X</math> in <math>\mathfrak g</math> such that
:<math>[H,X]=\alpha(H)X</math>
for all <math>H</math> in <math>\mathfrak h</math>. The collection of roots forms a [[root system]].

From the perspective of representation theory, the significance of the roots and root vectors is the following elementary but important result: If <math>\sigma : \mathfrak{g} \to \operatorname{End}(V)</math> is a representation of <math>\mathfrak g</math>, ''v'' is a weight vector with weight <math>\lambda</math> and ''X'' is a root vector with root <math>\alpha</math>, then
: <math>\sigma(H)(\sigma(X)(v))=[(\lambda+\alpha)(H)](\sigma(X)(v))</math>
for all ''H'' in <math>\mathfrak h</math>. That is, <math>\sigma(X)(v)</math> is either the zero vector or a weight vector with weight <math>\lambda+\alpha</math>. Thus, the action of <math>X</math> maps the weight space with weight <math>\lambda</math> into the weight space with weight <math>\lambda+\alpha</math>.

For example, if <math>\mathfrak{g}=\mathfrak{su}_{\mathbb{C}}(2)</math>, or <math>\mathfrak{su}(2)</math> complexified, the root vectors <math>{H,X,Y}</math> span the algebra and have weights <math>0</math>, <math>1</math>, and <math>-1</math> respectively. The Cartan subalgebra is spanned by <math>H</math>, and the action of <math>H</math> classifies the weight spaces. The action of <math>X</math> maps a weight space of weight <math>\lambda</math> to the weight space of weight <math>\lambda+1</math> and the action of <math>Y</math> maps a weight space of weight <math>\lambda</math> to the weight space of weight <math>\lambda-1</math>, and the action of <math>H</math> maps the weight spaces to themselves. In the fundamental representation, with weights <math>\pm\frac{1}{2}</math> and weight spaces <math>V_{\pm\frac{1}{2}}</math>, <math>X</math> maps <math>V_{+\frac{1}{2}}</math> to zero and <math>V_{-\frac{1}{2}}</math> to <math>V_{+\frac{1}{2}}</math>, while <math>Y</math> maps <math>V_{-\frac{1}{2}}</math> to zero and <math>V_{+\frac{1}{2}}</math> to <math>V_{-\frac{1}{2}}</math>, and <math>H</math> maps each weight space to itself.