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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

### Weight space of a representation

## Semisimple Lie algebras

### Ordering on the space of weights

### Integral weight

### Dominant weight

### Highest weight

### Highest weight module

### Verma module

## References

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 v ∈ V, 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: G → F^{×} 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.

This 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

- $chi(ab)\; =\; chi(a)chi(b)$ for all a, b in A.

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: 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.)

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: g → F is the subspace

- $V\_lambda:=\{vin\; V:\; forall\; xiin\; mathfrak\{g\},quad\; xicdot\; v=lambda(xi)v\}.$

If V is the direct sum of its weight spaces

- $V=bigoplus\_\{lambdainmathfrak\{g\}^*\}\; V\_lambda$

Similarly, we can define a weight space V_{λ} for any representation of a Lie group or an associative algebra.

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.

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

- μ ≤ λ if and only if λ − μ is nonnegative linear combination of simple roots.

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

- μ ≤ λ if and only if μ(f) ≤ λ(f).

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.

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.

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.

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.

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.

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Last updated on Friday September 26, 2008 at 13:29:46 PDT (GMT -0700)

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Last updated on Friday September 26, 2008 at 13:29:46 PDT (GMT -0700)

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