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# 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

$chi\left(gh\right) = chi\left(g\right)chi\left(h\right)$ 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

$chi\left(ab\right) = chi\left(a\right)chi\left(b\right)$ 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

$V_lambda:=\left\{vin V: forall xiin mathfrak\left\{g\right\},quad xicdot v=lambda\left(xi\right)v\right\}.$
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

$V=bigoplus_\left\{lambdainmathfrak\left\{g\right\}^*\right\} V_lambda$
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 $mathfrak\left\{h\right\}_0^*$ be the real subspace of $mathfrak\left\{h\right\}^*$ (if it is complex) generated by the roots of $mathfrak\left\{g\right\}$.

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

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 $finmathfrak\left\{h\right\}_0$ 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 $lambdainmathfrak\left\{h\right\}^*$ is integral (or $mathfrak\left\{g\right\}$-integral), if $lambda\left(H_gamma\right)inZ$ 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 $mathfrak\left\{h\right\}^*$ dual to the set of simple coroots $H_\left\{alpha_1\right\}, ldots, H_\left\{alpha_n\right\}$.

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

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

### Dominant weight

A weight λ is dominant, if $lambda\left(H_gamma\right)geq 0$ 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 $mathfrak\left\{g\right\}$ 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 $mathfrak\left\{g\right\}$.

Note that this is something more special than a $mathfrak\left\{g\right\}$-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 $lambdainmathfrak\left\{h\right\}^*$, there exists a unique (up to isomorphism) simple highest weight $mathfrak\left\{g\right\}$-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.

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