Weight (representation theory)


In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation 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 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 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 = 1 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.
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:
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.
If A is a Lie 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 = 0. A weight on a Lie algebra g over a field F is a linear map λ: g → F with λ = 0 for all x, y in g. Any weight on a Lie algebra g vanishes on the derived algebra and hence descends to a weight on the abelian Lie algebra g/. 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.

Weights in the representation theory of semisimple Lie algebras

Let be a complex semisimple Lie algebra and a Cartan subalgebra of. 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." The representations themselves are described in the article linked to above.

Weight of a representation

Let be a representation of a Lie algebra on a vector space V over a field of characteristic 0, say, and let be a linear functional on, where is a Cartan subalgebra of. Then the of V with weight λ is the subspace given by
A weight of the representation V 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, with the corresponding eigenvalues given by λ.
If V is the direct sum of its weight spaces
then V is called a ; this corresponds to there being a common eigenbasis for all the represented elements of the algebra, i.e., to there being simultaneously diagonalizable matrices.
If G is group with Lie algebra, every finite-dimensional representation of G induces a representation of. A weight of the representation of G is then simply a weight of the associated representation of. 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.

Action of the root vectors

For the adjoint representation of , the space over which the representation acts is the Lie algebra itself. Then the nonzero weights are called roots, the weight spaces are called root spaces, and the weight vectors, which are thus elements of, are called root vectors. Explicitly, a linear functional on the Cartan subalgebra is called a root if and there exists a nonzero in such that
for all in. 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 is a representation of, v is a weight vector with weight and X is a root vector with root, then
for all H in. That is, is either the zero vector or a weight vector with weight. Thus, the action of maps the weight space with weight into the weight space with weight.
For example, if, or complexified, the root vectors span the algebra and have weights,, and respectively. The Cartan subalgebra is spanned by, and the action of classifies the weight spaces. The action of maps a weight space of weight to the weight space of weight and the action of maps a weight space of weight to the weight space of weight, and the action of maps the weight spaces to themselves. In the fundamental representation, with weights and weight spaces, maps to zero and to, while maps to zero and to, and maps each weight space to itself.

Integral element

Let be the real subspace of generated by the roots of, where is the space of linear functionals, the dual space to. 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. We may then use this inner product to identify with a subspace of. With this identification, the coroot associated to a root is given as
where denotes the inner product of vectors 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 angle bracket notation should not be confused with the inner product
We now define two different notions of integrality for elements of. 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.
An element is algebraically integral if
for all roots. The motivation for this condition is that the coroot can be identified with the H element in a standard basis for an -subalgebra of. By elementary results for, the eigenvalues of in any finite-dimensional representation must be an integer. We conclude that, as stated above, the weight of any finite-dimensional representation of is algebraically integral.
The fundamental weights are defined by the property that they form a basis of dual to the set of coroots associated to the simple roots. That is, the fundamental weights are defined by the condition
where are the simple roots. An element is then algebraically integral if and only if it is an integral combination of the fundamental weights. The set of all -integral weights is a lattice in called the weight lattice for, denoted by.
The figure shows the example of the Lie algebra, whose root system is the root system. There are two simple roots, and. The first fundamental weight,, should be orthogonal to and should project orthogonally to half of, and similarly for. The weight lattice is then the triangular lattice.
Suppose now that the Lie algebra is the Lie algebra of a Lie group G. Then we say that is analytically integral if for each t in such that we have. The reason for making this definition is that if a representation of arises from a representation of G, then the weights of the representation will be G-integral. For G semisimple, the set of all G-integral weights is a sublattice P ⊂ P. If G is simply connected, then P = P. If G is not simply connected, then the lattice P is smaller than P and their quotient is isomorphic to the fundamental group of G.

Partial ordering on the space of weights

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. Recall that R is the set of roots; we now fix a set of positive roots.
Consider two elements and of . We are mainly interested in the case where and are integral, but this assumption is not necessary to the definition we are about to introduce. We then say that is higher than, which we write as, if is expressible as a linear combination of positive roots with non-negative real coefficients. This means, roughly, that "higher" means in the directions of the positive roots. We equivalently say that is "lower" than, which we write as.
This is only a partial ordering; it can easily happen that is neither higher nor lower than.