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 formula_1 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
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 (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. A weight on a Lie algebra g over a field F is a linear map λ: g → F with λ(["x", "y"])=0 for all "x", "y" in g. Any weight on a Lie algebra g vanishes on the derived algebra [g,g] and hence descends to a weight on the abelian Lie algebra g/[g,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. (For Lie groups, this is differentiation at the identity element of "G", and the algebraic group case is an abstraction using the notion of a derivation.)
Weights in the representation theory of semisimple Lie algebras.
Let formula_4 be a complex semisimple Lie algebra and formula_5 a Cartan subalgebra of formula_4. In this section, we describe the concepts needed to formulate the "theorem of the highest weight" classifying the finite-dimensional representations of formula_4. 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 "V" be a representation of a Lie algebra formula_4 over C and let λ be a linear functional on formula_5. Then the of "V" with weight λ is the subspace formula_10 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 formula_5, with the corresponding eigenvalues given by λ.
If "V" is the direct sum of its weight spaces
then it is called a ";" this corresponds to there being a common eigenbasis (a basis of simultaneous eigenvectors) for all the represented elements of the algebra, i.e., to their being simultaneously diagonalizable matrices (see diagonalizable matrix).
If "G" is group with Lie algebra formula_4, every finite-dimensional representation of "G" induces a representation of formula_4. A weight of the representation of "G" is then simply a weight of the associated representation of formula_4. 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.)
Action of the root vectors.
If "V" is the adjoint representation of formula_4, the nonzero weights of "V" are called roots, the weight spaces are called root spaces, and weight vectors are called root vectors. Explicitly, a linear functional formula_18 on formula_5 is called a root if formula_20 and there exists a nonzero formula_21 in formula_4 such that
for all formula_24 in formula_5. 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 "V" is a representation of formula_4, "v" is a weight vector with weight formula_27 and "X" is a root vector with root formula_18, then
for all "H" in formula_5. That is, formula_31 is either the zero vector or a weight vector with weight formula_32. Thus, the action of formula_21 maps the weight space with weight formula_27 into the weight space with weight formula_32.
Integral element.
Let formula_36 be the real subspace of formula_37 generated by the roots of formula_4. 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 formula_36 with a subspace formula_40 of formula_5. With this identification, the coroot associated to a root formula_18 is given as 
We now define two different notions of integrality for elements of formula_40. The motivation for these definitions is simple: The weights of finite-dimensional representations of formula_4 satisfy the first integrality condition, while if "G" is a group with Lie algebra formula_4, the weights of finite-dimensional representations of "G" satisfy the second integrality condition.
An element formula_47 is algebraically integral if
for all roots formula_18. The motivation for this condition is that the coroot formula_50 can be identified with the "H" element in a standard formula_51 basis for an sl(2,C)-subalgebra of g. By elementary results for sl(2,C), the eigenvalues of formula_50 in any finite-dimensional representation must be an integer. We conclude that, as stated above, the weight of any finite-dimensional representation of formula_4 is algebraically integral.
The fundamental weights formula_54 are defined by the property that they form a basis of formula_40 dual to the set of coroots associated to the simple roots. That is, the fundamental weights are defined by the condition
where formula_57 are the simple roots. An element formula_27 is then algebraically integral if and only if it is an integral combination of the fundamental weights. The set of all formula_4-integral weights is a lattice in formula_40 called "weight lattice" for formula_4, denoted by formula_62.
The figure shows the example of the Lie algebra sl(3,C), whose root system is the formula_63 root system. There are two simple roots, formula_64 and formula_65. The first fundamental weight, formula_66, should be orthogonal to formula_65 and should project orthogonally to half of formula_64, and similarly for formula_69. The weight lattice is then the triangular lattice.
Suppose now that the Lie algebra formula_4 is the Lie algebra of a Lie group "G". Then we say that formula_47 is analytically integral ("G-integral") if for each "t" in formula_5 such that formula_73 we have formula_74. The reason for making this definition is that if a representation of formula_4 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"("G") ⊂ "P"(formula_4). If "G" is simply connected, then "P"("G") = "P"(formula_4). If "G" is not simply connected, then the lattice "P"("G") is smaller than "P"(formula_4) 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 g. Recall that "R" is the set of roots; we now fix a set formula_79 of positive roots.
Consider two elements formula_80 and formula_27 of formula_40. We are mainly interested in the case where formula_80 and formula_27 are integral, but this assumption is not necessary to the definition we are about to introduce. We then say that formula_80 is higher than formula_27, which we write as formula_87, if formula_88 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 formula_27 is "lower" than formula_80, which we write as formula_91.
This is only a "partial" ordering; it can easily happen that formula_80 is neither higher nor lower than formula_27.
Dominant weight.
An integral element λ is "dominant" if formula_94 for each positive root "γ". Equivalently, λ is dominant if it is a "non-negative" integer combination of the fundamental weights. In the formula_63 case, the dominant integral elements live in a 60-degree sector. The notion of being dominant is not the same as being higher than zero.
The set of all λ (not necessarily integral) such that formula_94 is known as the "fundamental Weyl chamber" associated to the given set of positive roots.
Theorem of the highest weight.
A weight formula_27 of a representation formula_98 of formula_4 is called a highest weight if every other weight of formula_98 is lower than formula_27.
The theory classifying the finite-dimensional irreducible representations of formula_4 is by means of a "theorem of the highest weight." The theorem says that
The last point is the most difficult one; the representations may be constructed using Verma modules.
Highest-weight module.
A representation (not necessarily finite dimensional) "V" of formula_4 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 formula_4. Every irreducible formula_4-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. 
For each formula_106—not necessarily dominant or integral—there exists a unique (up to isomorphism) simple highest-weight formula_4-module with highest weight λ, which is denoted "L"(λ), but this module is infinite dimensional unless λ is dominant integral. 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.
Every "finite-dimensional" highest weight module is irreducible.