Weight (representation theory)

In the mathematical field of representation theory, the concept of weights of an algebra A over a field F is a generalisation of that of eigenvalues. They are algebra homomorphisms from A to F, or equivalently, one-dimensional representations of A over F. It is the algebra analogue of a multiplicative character of a group. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space.

Motivation and general concept

Given a set S of

n

×

n

{\displaystyle n\times n}

matrices over the same field, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V defines a linear functional on the subalgebra U of End(V ) generated by the set of endomorphisms S; this functional is defined as the map which associates to each element of U its eigenvalue on the eigenvector v. This map is also multiplicative, and sends the identity to 1; thus it is an algebra homomorphism from U to the base field. This "generalized eigenvalue" is a prototype for the notion of a weight.

The notion is closely related to the idea of a multiplicative character in group theory, which is a homomorphism χ from a group G to the multiplicative group of a field F. Thus χ: G → F× satisfies χ(e) = 1 (where e is the identity element of G) and

χ

(

g

h

)

=

χ

(

g

)

χ

(

h

)

for all

g

,

h

∈

G

.

{\displaystyle \chi (gh)=\chi (g)\chi (h)\ {\text{ for all }}g,h\in G.}

Indeed, if G acts on a vector space V over F, each simultaneous eigenspace for every element of G, if such exists, determines a multiplicative character on G: the eigenvalue on this common eigenspace of each element of the group.

The notion of multiplicative character can be extended to any algebra A over F, by replacing χ: G → F× by a linear map χ: A → F with:

χ

(

a

b

)

=

χ

(

a

)

χ

(

b

)

{\displaystyle \chi (ab)=\chi (a)\chi (b)}

for all a, b in A. If an algebra A acts on a vector space V over F to any simultaneous eigenspace, this corresponds an algebra homomorphism from A to F assigning to each element of A its eigenvalue.