# 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 simultaneously diagonalize all of the elements of *S*.^{[note 1]}^{[note 2]} 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

- for all
*g*,*h*in*G*.

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.)

### Weight space of a representation of a Lie algebra

Among the set of weights, some are related to the data 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: **ħ** → **F** (**ħ** is the Cartan subalgebra of **g**.) is the subspace

(where denotes the action of **ħ** on *V*). A *weight of the representation* *V* is 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

then it is called a *weight module;* 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).

Similarly, we can define a weight space *V*_{λ} 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 Φ^{+} has been fixed. This is equivalent to the choice of a set of simple roots.

### Ordering on the space of weights

Let **h***_{0} be the real subspace of **h*** (if it is complex) generated by the roots of **g**.

There are two concepts how to define an ordering of **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 *f* in **h**_{0} and

- μ ≤ λ 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 λ ∈ **h*** is *integral* (or **g**-integral), if λ(*H*_{γ}) ∈ **Z** for each coroot *H*_{γ} such that γ is a positive root.

The fundamental weights are defined by the property that they form a basis of **h*** dual to the set of simple coroots .

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

A weight λ of the Lie group *G* is called integral, if for each *t* in **h** such that . For *G* semisimple, the set of all *G*-integral weights is a sublattice *P*(*G*) ⊂ *P*(**g**). If *G* is simply connected, then *P*(*G*) = *P*(**g**). If *G* is not simply connected, then the lattice *P*(*G*) is smaller than *P*(**g**) and their quotient is isomorphic to the fundamental group of *G*.^{[1]}

### Dominant weight

A weight λ is *dominant* if for each coroot *H*_{γ} 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 a *highest weight* if no other weight of *V* is larger than λ in the partial order given above. Sometimes, one imposes the stronger condition that all other weights of *V* are strictly smaller than λ in the partial order. The term *highest weight* often suggests (or denotes) the highest weight of a "highest-weight module".

One defines a *lowest weight* similarly.

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 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 **g** 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 **g**. Every finite-dimensional, irreducible representation of a semisimple Lie algebra *g* is a highest-weight module, and the representations can be classified by their highest weights ("theorem of the highest weight").^{[2]}

This is something more special than a **g**-module with a highest weight.

Similarly we can define a highest-weight module for representation of a Lie group.

### Verma module

For each dominant weight λ ∈ **h***, there exists a unique (up to isomorphism) simple highest-weight **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 module is a weight module. The weight spaces in a highest-weight module are always finite dimensional.

## See also

## Notes

- ↑ The converse is also true – a set of diagonalizable matrices commutes if and only if the set is simultaneously diagonalisable (Horn & Johnson 1985, pp. 51–53).
- ↑ In fact, given a set of commuting matrices over an algebraically closed field, they are simultaneously triangularizable, without needing to assume that they are diagonalizable.

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