Weight (representation theory)
In the
Motivation and general concept
Given a set S of
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
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.)
Weights in the representation theory of semisimple Lie algebras
Let be a complex semisimple Lie algebra and a Cartan subalgebra of . In this section, we describe the concepts needed to formulate the "theorem of the highest weight" classifying the finite-dimensional representations of . Notably, we will explain the notion of a "dominant integral element." The representations themselves are described in the article linked to above.
Weight of a representation
Let be a representation of a Lie algebra on a vector space V over a field of characteristic 0, say , and let be a linear functional on . Then the weight space of V with weight λ is the subspace given by
- .
A weight of the representation V (the representation is often referred to in short by the vector space V over which elements of the Lie algebra act rather than the map ) is a linear functional λ such that the corresponding weight space is nonzero. Nonzero elements of the weight space are called weight vectors. That is to say, a weight vector is a simultaneous eigenvector for the action of the elements of , with the corresponding eigenvalues given by λ.
If V is the direct sum of its weight spaces
then V is called a weight module; this corresponds to there being a common
If G is group with Lie algebra , every finite-dimensional representation of G induces a representation of . A weight of the representation of G is then simply a weight of the associated representation of . There is a subtle distinction between weights of group representations and Lie algebra representations, which is that there is a different notion of integrality condition in the two cases; see below. (The integrality condition is more restrictive in the group case, reflecting that not every representation of the Lie algebra comes from a representation of the group.)
Action of the root vectors
For the
for all in . The collection of roots forms a root system.
From the perspective of representation theory, the significance of the roots and root vectors is the following elementary but important result: If is a representation of , v is a weight vector with weight and X is a root vector with root , then
for all H in . That is, is either the zero vector or a weight vector with weight . Thus, the action of maps the weight space with weight into the weight space with weight .
For example, if , or complexified, the root vectors span the algebra and have weights , , and respectively. The Cartan subalgebra is spanned by , and the action of classifies the weight spaces. The action of maps a weight space of weight to the weight space of weight and the action of maps a weight space of weight to the weight space of weight , and the action of maps the weight spaces to themselves. In the fundamental representation, with weights and weight spaces , maps to zero and to , while maps to zero and to , and maps each weight space to itself.
Integral element
Let be the real subspace of generated by the roots of , where is the space of linear functionals , the dual space to . For computations, it is convenient to choose an inner product that is invariant under the Weyl group, that is, under reflections about the hyperplanes orthogonal to the roots. We may then use this inner product to identify with a subspace of . With this identification, the
where denotes the
We now define two different notions of integrality for elements of . The motivation for these definitions is simple: The weights of finite-dimensional representations of satisfy the first integrality condition, while if G is a group with Lie algebra , the weights of finite-dimensional representations of G satisfy the second integrality condition.
An element is algebraically integral if
for all roots . The motivation for this condition is that the coroot can be identified with the H element in a standard basis for an -subalgebra of .[1] By elementary results for , the eigenvalues of in any finite-dimensional representation must be an integer. We conclude that, as stated above, the weight of any finite-dimensional representation of is algebraically integral.[2]
The fundamental weights are defined by the property that they form a basis of dual to the set of coroots associated to the simple roots. That is, the fundamental weights are defined by the condition
where are the simple roots. An element is then algebraically integral if and only if it is an integral combination of the fundamental weights.[3] The set of all -integral weights is a lattice in called the weight lattice for , denoted by .
The figure shows the example of the Lie algebra , whose root system is the root system. There are two simple roots, and . The first fundamental weight, , should be orthogonal to and should project orthogonally to half of , and similarly for . The weight lattice is then the triangular lattice.
Suppose now that the Lie algebra is the Lie algebra of a Lie group G. Then we say that is analytically integral (G-integral) if for each t in such that we have . The reason for making this definition is that if a representation of arises from a representation of G, then the weights of the representation will be G-integral.[4] For G semisimple, the set of all G-integral weights is a sublattice P(G) ⊂ P(). If G is
Partial ordering on the space of weights
We now introduce a partial ordering on the set of weights, which will be used to formulate the theorem of the highest weight describing the representations of . Recall that R is the set of roots; we now fix a set of positive roots.
Consider two elements and of . We are mainly interested in the case where and are integral, but this assumption is not necessary to the definition we are about to introduce. We then say that is higher than , which we write as , if is expressible as a linear combination of positive roots with non-negative real coefficients.[6] This means, roughly, that "higher" means in the directions of the positive roots. We equivalently say that is "lower" than , which we write as .
This is only a partial ordering; it can easily happen that is neither higher nor lower than .
Dominant weight
An integral element λ is dominant if for each positive root γ. Equivalently, λ is dominant if it is a non-negative integer combination of the fundamental weights. In the case, the dominant integral elements live in a 60-degree sector. The notion of being dominant is not the same as being higher than zero. Note the grey area in the picture on the right is a 120-degree sector, strictly containing the 60-degree sector corresponding to the dominant integral elements.
The set of all λ (not necessarily integral) such that is known as the fundamental Weyl chamber associated to the given set of positive roots.
Theorem of the highest weight
A weight of a representation of is called a highest weight if every other weight of is lower than .
The theory classifying the finite-dimensional irreducible representations of is by means of a "theorem of the highest weight." The theorem says that[7]
- (1) every irreducible (finite-dimensional) representation has a highest weight,
- (2) the highest weight is always a dominant, algebraically integral element,
- (3) two irreducible representations with the same highest weight are isomorphic, and
- (4) every dominant, algebraically integral element is the highest weight of an irreducible representation.
The last point is the most difficult one; the representations may be constructed using Verma modules.
Highest-weight module
A representation (not necessarily finite dimensional) V of is called highest-weight module if it is generated by a weight vector v ∈ V that is annihilated by the action of all
Every finite-dimensional highest weight module is irreducible.[8]
See also
- Classifying finite-dimensional representations of Lie algebras
- Representation theory of a connected compact Lie group
- Highest-weight category
- Root system
Notes
- simultaneously triangularizable, without needing to assume that they are diagonalizable.
References
- ^ Hall 2015 Theorem 7.19 and Eq. (7.9)
- ^ Hall 2015 Proposition 9.2
- ^ Hall 2015 Proposition 8.36
- ^ Hall 2015 Proposition 12.5
- ^ Hall 2015 Corollary 13.8 and Corollary 13.20
- ^ Hall 2015 Definition 8.39
- ^ Hall 2015 Theorems 9.4 and 9.5
- ^ This follows from (the proof of) Proposition 6.13 in Hall 2015 together with the general result on complete reducibility of finite-dimensional representations of semisimple Lie algebras
- OCLC 246650103..
- Goodman, Roe; Wallach, Nolan R. (1998), Representations and Invariants of the Classical Groups, Cambridge University Press, ISBN 978-0-521-66348-9.
- Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666
- Humphreys, James E. (1972a), Introduction to Lie Algebras and Representation Theory, Birkhäuser, ISBN 978-0-387-90053-7.
- Humphreys, James E. (1972b), Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 21, Berlin, New York: MR 0396773
- Knapp, Anthony W. (2002), Lie Groups Beyond an Introduction (2nd ed.), Birkhäuser, ISBN 978-0-8176-4259-4.