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 – a linear functional – 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.

1 Motivation and general concept
- 1.1 Weights
- 1.2 Weight space of a representation
2 Semisimple Lie algebras
3 See also
4 Notes
5 References

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 "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.

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

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

If an algebra A acts on a vector space V over F to any simultaneous eigenspace corresponds an 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 this map vanish on commutators : 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

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

$V_\lambda:=\{v\in V: \forall \xi\in \mathfrak{g},\quad \xi\cdot v=\lambda(\xi)v\}.$

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

$V=\bigoplus_{\lambda\in\mathfrak{g}^*} V_\lambda$

then it is called a weight module; this corresponds to having an eigenbasis (a basis of eigenvectors), i.e., being a 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 $\Phi^%2B$ 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

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

The second concept is given by an element $f\in\mathfrak{h}_0$ 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 $\lambda\in\mathfrak{h}^*$ is integral (or $\mathfrak{g}$ -integral), if $\lambda(H_\gamma)\in\Z$ for each coroot $H_\gamma$ such that $\gamma$ is a positive root.

The fundamental weights $\omega_1,\ldots,\omega_n$ 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 $t\in\mathfrak{h}$ such that $\exp(t)=1\in G,\,\,\lambda(t)\in 2\pi i \mathbb{Z}$ . 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 $v\in V$ that is annihilated by the action of all positive root spaces in $\mathfrak{g}$ .

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

Main article: Verma module

For each weight $\lambda\in\mathfrak{h}^*$ , 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 module is a weight module. The weight spaces in a highest-weight module are always finite dimensional.

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.

References

Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics, 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN 978-0-387-97527-6 .
Goodman, Roe; Wallach, Nolan R. (1998), Representations and Invariants of the Classical Groups, Cambridge University Press, ISBN 978-0521663489 .
Humphreys, James E. (1972a), Introduction to Lie Algebras and Representation Theory, Birkhäuser, ISBN 978-0387900537 .
Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6
Humphreys, James E. (1972b), Linear Algebraic Groups, Graduate Texts in Mathematics, 21, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90108-4, MR 0396773
Knapp, Anthony W. (2002), Lie Groups Beyond an Introduction (2nd ed.), Birkhäuser, ISBN 978-0817642594 .