Editing Weight (representation theory) (section)

===Weight of a representation===
[[File:A2example.pdf|thumb|Example of the weights of a representation of the Lie algebra sl(3,C)]]
Let <math>\sigma : \mathfrak{g} \to \operatorname{End}(V)</math> be a representation of a Lie algebra <math>\mathfrak g</math> on a vector space ''V'' over a field of characteristic 0, say <math>\mathbb{C}</math>, and let <math>\lambda : \mathfrak{h} \to \mathbb{C}</math> be a linear functional on <math>\mathfrak h</math>, where <math>\mathfrak h</math> is a [[Cartan subalgebra]] of <math>\mathfrak g</math>. Then the '''{{visible anchor|weight space}}''' of ''V'' with weight ''λ'' is the subspace <math>V_\lambda</math> given by
:<math>V_\lambda:=\{v\in V: \forall H\in \mathfrak{h},\, (\sigma(H))(v)=\lambda(H)v\}</math>.
A '''weight''' of the representation ''V'' (the representation is often referred to in short by the vector space ''V'' over which elements of the Lie algebra act rather than the map <math>\sigma</math>) is a linear functional λ such that the corresponding weight space is nonzero. Nonzero elements of the weight space are called '''weight vectors'''. That is to say, a weight vector is a simultaneous eigenvector for the action of the elements of <math>\mathfrak h</math>, with the corresponding eigenvalues given by λ.

If ''V'' is the direct sum of its weight spaces
:<math>V=\bigoplus_{\lambda\in\mathfrak{h}^*} V_\lambda</math>
then ''V'' is called a ''{{visible anchor|weight module}};'' this corresponds to there being a common [[eigenbasis]] (a basis of simultaneous eigenvectors) for all the represented elements of the algebra, i.e.,  to there being simultaneously diagonalizable matrices (see [[diagonalizable matrix]]).

If ''G'' is group with Lie algebra <math>\mathfrak g</math>, every finite-dimensional representation of ''G'' induces a representation of <math>\mathfrak g</math>. A weight of the representation of ''G'' is then simply a weight of the associated representation of <math>\mathfrak g</math>. There is a subtle distinction between weights of group representations and Lie algebra representations, which is that there is a different notion of integrality condition in the two cases; see below. (The integrality condition is more restrictive in the group case, reflecting that not every representation of the Lie algebra comes from a representation of the group.)