Weight (representation theory)
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.[note 1] 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 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.
If A is a Lie algebra (which is generally not an associative algebra), then instead of requiring multiplicativity of a character, one requires that it maps any Lie bracket to the corresponding commutator; but since F is commutative this simply means that this map must vanish on Lie brackets: χ([a,b]) = 0.
Thus weights are primarily of interest for abelian Lie algebras, where they reduce to the simple notion of a generalized eigenvalue for space of commuting linear transformations.
If G is a Lie group or an algebraic group, then a multiplicative character θ: G → F× induces a weight χ = dθ: g → F on its Lie algebra by differentiation.
In this section, we describe the concepts needed to formulate the "theorem of the highest weight" classifying the finite-dimensional representations of
Notably, we will explain the notion of a "dominant integral element."
) is a linear functional λ such that the corresponding weight space is nonzero.
That is to say, a weight vector is a simultaneous eigenvector for the action of the elements of
If V is the direct sum of its weight spaces then V is called a 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).
For computations, it is convenient to choose an inner product that is invariant under the Weyl group, that is, under reflections about the hyperplanes orthogonal to the roots.
In addition to this inner product, it is common for an angle bracket notation
to be used in discussions of root systems, with the angle bracket defined as
The angle bracket here is not an inner product, as it is not symmetric, and is linear only in the first argument.
The motivation for these definitions is simple: The weights of finite-dimensional representations of
satisfy the first integrality condition, while if G is a group with Lie algebra
, the weights of finite-dimensional representations of G satisfy the second integrality condition.
dual to the set of coroots associated to the simple roots.
) and their quotient is isomorphic to the fundamental group of G.[5] We now introduce a partial ordering on the set of weights, which will be used to formulate the theorem of the highest weight describing the representations of
is expressible as a linear combination of positive roots with non-negative real coefficients.
for each positive root γ. Equivalently, λ is dominant if it is a non-negative integer combination of the fundamental weights.
case, the dominant integral elements live in a 60-degree sector.
is known as the fundamental Weyl chamber associated to the given set of positive roots.
The theorem says that[7] The last point is the most difficult one; the representations may be constructed using Verma modules.
is called highest-weight module if it is generated by a weight vector v ∈ V that is annihilated by the action of all positive root spaces in
-module with a highest weight is necessarily a highest-weight module, but in the infinite-dimensional case, a highest weight module need not be irreducible.
—not necessarily dominant or integral—there exists a unique (up to isomorphism) simple highest-weight
It can be shown that each highest weight module with highest weight λ is a quotient of the Verma module M(λ).
This is just a restatement of universality property in the definition of a Verma module.
