Weight (representation theory)

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

Weights

Given a set S of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to diagonalize all of the elements of S simultaneously. 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. The set S spans a linear subspace U of End(V) and for each eigenvector vV, there is linear functional on U which associates to each element of U its eigenvalue on the eigenvector v. 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: GF× satisfies χ(e) = 1 (where e is the identity element of G) and

chi(gh) = chi(g)chi(h) for all g, h in G.
If G acts on a vector space V over F and there is a simultaneous eigenspace for every element of G, then the eigenvalue of each element determines a multiplicative character on G.

This notion of multiplicative character can be extended to any algebra A over F, by replacing χ: GF× by a linear map χ: AF with

chi(ab) = chi(a)chi(b) for all a, b in A.
If an algebra A acts on a vector space V over F and there is a simultaneous eigenspace, then there is a corresponding algebra homomorphism from A to F assigning to each element of A its eigenvalue.

If A is a Lie algebra, then the commutativity of the field and the anticommutativity of the Lie bracket imply that χ([a,b])=0. A weight on a Lie algebra g over a field F is a linear map λ: gF 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: GF× induces a weight χ = dθ: gF 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.)

Weight space of a representation

Let V be a representation of a Lie algebra g over a field F and let λ be a weight of g. Then the weight space of V with weight λ: gF is the subspace

V_lambda:={vin V: forall xiin mathfrak{g},quad xicdot v=lambda(xi)v}.
A weight of the representation V is a a weight λ such that the corresponding weight space is nonzero. Nonzero elements of the weight space are called weight vectors.

If V is the direct sum of its weight spaces

V=bigoplus_{lambdainmathfrak{g}^*} V_lambda
then it is called a called a weight module.

Similarly, we can define a weight space Vλ for any representation of a Lie group or an associative algebra.

Semisimple Lie algebras

Let g be a Lie algebra, h a maximal commutative Lie subalgebra consisting of semi-simple elements (sometimes called Cartan subalgebra) and let V be a finite dimensional representation of g. If g is semisimple, then [g,g] = g and so all weights on g are trivial. However, V is, by restriction, a representation of h, and it is well known that V is a weight module for h, i.e., equal to the direct sum of its weight spaces. By an abuse of language, the weights of V as a representation of h are often called weights of V as a representation of g.

Similar definitions apply to a Lie group G, a maximal commutative Lie subgroup H and any representation V of G. Clearly, if λ is a weight of the representation V of G, it is also a weight of V as a representation of the Lie algebra g of G.

If V is the adjoint representation of g, its weights are called roots, the weight spaces are called root spaces, and weight vectors are sometimes called root vectors.

We now assume that g is semisimple, with a chosen Cartan subalgebra h and corresponding root system. Let us suppose also that a choice of positive roots Phi^+ has been fixed. This is equivalent to the choice of a set of simple roots.

Ordering on the space of weights

Let mathfrak{h}_0^* be the real subspace of mathfrak{h}^* (if it is complex) generated by the roots of mathfrak{g}.

There are two concepts how to define an ordering of mathfrak{h}_0^*.

The first one is

μλ if and only if λμ is nonnegative linear combination of simple roots.

The second concept is given by an element finmathfrak{h}_0 and

μλ if and only if μ(f) ≤ λ(f).
Usually, f is chosen so, that β(f) > 0 for each positive root β.

Integral weight

A weight lambdainmathfrak{h}^* is integral (or mathfrak{g}-integral), if lambda(H_gamma)inZ for each coroot H_gamma such that gamma is a positive root.

The fundamental weights varpi_1,ldots,varpi_n are defined by the property that they form a basis of mathfrak{h}^* dual to the set of simple coroots H_{alpha_1}, ldots, H_{alpha_n}.

Hence λ is integral if it is an integral combination of the fundamental weights. The set of all mathfrak{g}-integral weights is a lattice in mathfrak{h}^* called weight lattice for mathfrak{g}, denoted by P(mathfrak{g}).

A weight λ of the Lie group G is called integral, if for each tinmathfrak{h} such that exp(t)=1in G,,,lambda(t)in 2pi i mathbb{Z}. For G semisimple, the set of all G-integral weights is a sublattice P(G)subset P(mathfrak{g}). If G is simply connected, then P(G)=P(mathfrak{g}). If G is not simply connected, then the lattice P(G) is smaller than P(mathfrak{g}) and their quotient is isomorphic to the fundamental group of G.

Dominant weight

A weight λ is dominant, if lambda(H_gamma)geq 0 for each coroot H_gamma such that γ is a positive root. Equivalently, λ is dominant, if it is a non-negative linear combination of the fundamental weights.

The convex hull of the dominant weights is sometimes called the fundamental Weyl chamber.

Sometimes, the term dominant weight is used to denote a dominant (in the above sense) and integral weight.

Highest weight

A weight λ of a representation V is called highest weight, if no other weight of V is larger than λ. Sometimes, it is assumed that a highest weight is a weight, such that all other weights of V are strictly smaller than λ in the partial ordering given above. The term highest weight denotes often the highest weight of a "highest weight module".

Similarly, we define the lowest weight.

The space of all possible weights is a vector space. Let's fix a total ordering of this vector space such that a nonnegative linear combination of positive vectors with at least one nonzero coefficient is another positive vector.

Then, a representation is said to have highest weight λ if λ is a weight and all its other weights are less than λ.

Similarly, it is said to have lowest weight λ if λ is a weight and all its other weights are greater than it.

A weight vector v_lambda in V of weight λ is called a highest weight vector, or vector of highest weight, if all other weights of V are smaller than λ.

Highest weight module

A representation V of mathfrak{g} is called highest weight module, if it is generated by a weight vector vin V that is annihilated by the action of all positive root spaces in mathfrak{g}.

Note that this is something more special than a mathfrak{g}-module with a highest weight.

Similarly we can define a highest weight module for representation of a Lie group or an associative algebra.

Verma module

For each weight lambdainmathfrak{h}^*, there exists a unique (up to isomorphism) simple highest weight mathfrak{g}-module with highest weight λ, which is denoted L(λ).

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.

A highest weight modules is a weight module. The weight spaces in a highest weight module are always finite dimensional.

References

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