View source for Weight (representation theory)

In the [[Mathematics|mathematical]] field of [[Representation theory|representation theory]], a '''weight''' of an [[Algebra over a field|algebra]] ''A'' over a [[Field (mathematics)|field]] '''F''' is an [[Algebra homomorphism|algebra homomorphism]] from ''A'' to '''F''', or equivalently, a one-dimensional [[Algebra representation|representation]] of ''A'' over '''F'''. It is the algebra analogue of a [[Multiplicative character|multiplicative character]] of a [[Group (mathematics)|group]]. The importance of the concept, however, stems from its application to [[Lie algebra representation|representations of Lie algebras]] and hence also to [[Group representation|representations]] of [[Algebraic group|algebraic]] and [[Lie group]]s. 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==

Given a set ''S'' of <math>n \times n</math> [[Matrix (mathematics)|matrices]] over the same field, each of which is [[Diagonalizable matrix|diagonalizable]], and any two of which [[Commuting matrices|commute]], it is always possible to simultaneously diagonalize all of the elements of ''S''.<ref group="note">In fact, given a set of commuting matrices over an [[Algebraically closed field|algebraically closed field]], they are simultaneously triangularizable, without needing to assume that they are diagonalizable.</ref> Equivalently, for any set ''S'' of mutually commuting [[Semisimple operator|semisimple]] linear transformations of a [[Dimension (vector space)|finite-dimensional]] [[Vector space|vector space]] ''V'' there exists a [[Basis (linear algebra)|basis]] of ''V'' consisting of ''{{anchor|simultaneous eigenvector}}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''&thinsp;) generated by the set of [[Endomorphism|endomorphism]]s ''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|multiplicative character]] in [[Group theory|group theory]], which is a [[Group homomorphism|homomorphism]] ''χ'' from a group ''G'' to the [[Multiplicative group|multiplicative group]] of a field '''F'''. Thus ''χ'': ''G'' → '''F'''<sup>×</sup> satisfies ''χ''(''e'') = 1 (where ''e'' is the [[Identity element|identity element]] of ''G'') and
:<math>\chi(gh) = \chi(g)\chi(h)</math> for all ''g'', ''h'' in ''G''.
Indeed, if ''G'' [[Group representation|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'''<sup>×</sup> by a [[Linear map|linear map]] ''χ'': ''A'' → '''F''' with:
:<math>\chi(ab) = \chi(a)\chi(b)</math> 
for all ''a'', ''b'' in ''A''. If an algebra ''A'' [[Algebra representation|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|associative algebra]]), then instead of requiring multiplicativity of a character, one requires that it maps any Lie bracket to the corresponding [[Commutator|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|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|algebraic group]], then a multiplicative character θ: ''G'' → '''F'''<sup>×</sup> 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 <math>\mathfrak g</math> be a complex semisimple Lie algebra and <math>\mathfrak h</math> a Cartan subalgebra of <math>\mathfrak g</math>. In this section, we describe the concepts needed to formulate the "theorem of the highest weight" classifying the finite-dimensional representations of <math>\mathfrak g</math>. 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===
[[File:A2example.pdf|thumb|Example of the weights of a representation of the Lie algebra sl(3,C)]]
Let <math>\sigma : \mathfrak{g} \to End(V)</math> be a representation of a Lie algebra <math>\mathfrak g</math> on a vector space ''V'' over a field of characteristic 0, say <math>\mathbb{C}</math>, and let <math>\lambda : \mathfrak{h} \to \mathbb{C}</math> be a linear functional on <math>\mathfrak h</math>. Then the '''{{visible anchor|weight space}}''' of ''V'' with weight ''λ'' is the subspace <math>V_\lambda</math> given by
:<math>V_\lambda:=\{v\in V: \forall H\in \mathfrak{h},\quad (\sigma(H))(v)=\lambda(H)v\}</math>.
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 <math>\sigma</math>) 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 <math>\mathfrak h</math>, with the corresponding eigenvalues given by λ.

If ''V'' is the direct sum of its weight spaces
:<math>V=\bigoplus_{\lambda\in\mathfrak{h}^*} V_\lambda</math>
then ''V'' is called a ''{{visible anchor|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 there being simultaneously diagonalizable matrices (see [[Diagonalizable matrix|diagonalizable matrix]]).

If ''G'' is group with Lie algebra <math>\mathfrak g</math>, every finite-dimensional representation of ''G'' induces a representation of <math>\mathfrak g</math>. A weight of the representation of ''G'' is then simply a weight of the associated representation of <math>\mathfrak g</math>. 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 adjoint representation <math>\mathrm{ad} : \mathfrak{g}\to End(\mathfrak{g})</math> of  <math>\mathfrak g</math>, the space over which the representation acts is the Lie algebra itself. Then the nonzero weights are called '''[[Root system|roots]]''', the weight spaces are called '''root spaces''', and the weight vectors, which are thus elements of <math>\mathfrak{g}</math>, are called '''root vectors'''. Explicitly, a linear functional <math>\alpha</math> on <math>\mathfrak h</math> is called a root if <math>\alpha\neq 0</math> and there exists a nonzero <math>X</math> in <math>\mathfrak g</math> such that
:<math>[H,X]=\alpha(H)X</math>
for all <math>H</math> in <math>\mathfrak h</math>. The collection of roots forms a [[Root system|root system]].

From the perspective of representation theory, the significance of the roots and root vectors is the following elementary but important result: If <math>\sigma : \mathfrak{g} \to End(V)</math> is a representation of <math>\mathfrak g</math>, ''v'' is a weight vector with weight <math>\lambda</math> and ''X'' is a root vector with root <math>\alpha</math>, then
: <math>\sigma(H)(\sigma(X)(v))=[(\lambda+\alpha)(H)](\sigma(X)(v))</math>
for all ''H'' in <math>\mathfrak h</math>. That is, <math>\sigma(X)(v)</math> is either the zero vector or a weight vector with weight <math>\lambda+\alpha</math>. Thus, the action of <math>X</math> maps the weight space with weight <math>\lambda</math> into the weight space with weight <math>\lambda+\alpha</math>.

For example, if <math>\mathfrak{g}=\mathfrak{su}_{\mathbb{C}}(2)</math>, or <math>\mathfrak{su}(2)</math> complexified, the root vectors <math>{H,X,Y}</math> span the algebra and have weights <math>0</math>, <math>1</math>, and <math>-1</math> respectively. The Cartan subalgebra is spanned by <math>H</math>, and the action of <math>H</math> classifies the weight spaces. The action of <math>X</math> maps a weight space of weight <math>\lambda</math> to the weight space of weight <math>\lambda+1</math> and the action of <math>Y</math> maps a weight space of weight <math>\lambda</math> to the weight space of weight <math>\lambda-1</math>, and the action of <math>H</math> maps the weight spaces to themselves. In the fundamental representation, with weights <math>\pm\frac{1}{2}</math> and weight spaces <math>V_{\pm\frac{1}{2}}</math>, <math>X</math> maps <math>V_{+\frac{1}{2}}</math> to zero and <math>V_{-\frac{1}{2}}</math> to <math>V_{+\frac{1}{2}}</math>, while <math>Y</math> maps <math>V_{-\frac{1}{2}}</math> to zero and <math>V_{+\frac{1}{2}}</math> to <math>V_{-\frac{1}{2}}</math>, and <math>H</math> maps each weight space to itself.

===Integral element===
[[File:Weights_for_A2_root_system.png|thumb|right|Algebraically integral elements (triangular lattice), dominant integral elements (black dots), and fundamental weights for sl(3,C)]]
Let <math>\mathfrak h^*_0</math> be the real subspace of <math>\mathfrak h^*</math> generated by the roots of <math>\mathfrak g</math>, where <math>\mathfrak h^* </math> is the space of linear functionals <math>\lambda : \mathfrak h \to \mathbb C</math>, the dual space to <math>\mathfrak h</math>. 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 <math>\mathfrak h^*_0</math> with a subspace <math>\mathfrak h_0</math> of <math>\mathfrak h</math>. With this identification, the coroot associated to a root <math>\alpha</math> is given as 
:<math>H_\alpha=2\frac{\alpha}{\langle\alpha,\alpha\rangle}</math>.{{Definition needed|the angle bracket operator is not defined|date=July 2023}}

We now define two different notions of integrality for elements of <math>\mathfrak h_0</math>. The motivation for these definitions is simple: The weights of finite-dimensional representations of <math>\mathfrak g</math> satisfy the first integrality condition, while if ''G'' is a group with Lie algebra <math>\mathfrak g</math>, the weights of finite-dimensional representations of ''G'' satisfy the second integrality condition.

An element <math>\lambda\in\mathfrak h_0</math> is '''algebraically integral''' if
:<math>\langle\lambda,H_\alpha\rangle=2\frac{\langle\lambda,\alpha\rangle}{\langle\alpha,\alpha\rangle}\in\mathbb{Z}</math>
for all roots <math>\alpha</math>. The motivation for this condition is that the coroot <math>H_\alpha</math> can be identified with the ''H'' element in a standard <math>{X,Y,H}</math> basis for an <math>sl(2,\mathbb C)</math>-subalgebra of <math>\mathfrak g</math>.<ref>{{harvnb|Hall|2015}} Theorem 7.19 and Eq. (7.9)</ref> By elementary results for <math>sl(2,\mathbb C)</math>, the eigenvalues of <math>H_\alpha</math> in any finite-dimensional representation must be an integer. We conclude that, as stated above, the weight of any finite-dimensional representation of <math>\mathfrak g</math> is algebraically integral.<ref>{{harvnb|Hall|2015}} Proposition 9.2</ref>

The '''fundamental weights''' <math>\omega_1,\ldots,\omega_n</math> are defined by the property that they form a basis of <math>\mathfrak h_0</math> dual to the set of coroots associated to the [[Root system#Positive roots and simple roots|simple roots]]. That is, the fundamental weights are defined by the condition
:<math>2\frac{\langle\omega_i,\alpha_j\rangle}{\langle\alpha_j,\alpha_j\rangle}=\delta_{i,j}</math>
where  <math>\alpha_1,\ldots\alpha_n</math> are the simple roots. An element <math>\lambda</math> is then algebraically integral if and only if it is an integral combination of the fundamental weights.<ref>{{harvnb|Hall|2015}} Proposition 8.36</ref> The set of all <math>\mathfrak g</math>-integral weights is a [[Lattice (group)|lattice]] in <math>\mathfrak h_0</math> called the ''weight lattice'' for <math>\mathfrak g</math>, denoted by <math>P(\mathfrak g)</math>.

The figure shows the example of the Lie algebra <math>sl(3,\mathbb C)</math>, whose root system is the <math>A_2</math> root system. There are two simple roots, <math>\gamma_1</math> and <math>\gamma_2</math>. The first fundamental weight, <math>\omega_1</math>, should be orthogonal to <math>\gamma_2</math> and should project orthogonally to half of <math>\gamma_1</math>, and similarly for <math>\omega_2</math>. The weight lattice is then the triangular lattice.

Suppose now that the Lie algebra <math>\mathfrak g</math> is the Lie algebra of a Lie group ''G''. Then we say that <math>\lambda\in\mathfrak h_0</math> is '''analytically integral''' (''G-integral'') if for each ''t'' in <math>\mathfrak h</math> such that <math>\exp(t)=1\in G</math> we have <math>\langle\lambda,t\rangle\in 2\pi i \mathbb{Z}</math>. The reason for making this definition is that if a representation of <math>\mathfrak g</math> arises from a representation of ''G'', then the weights of the representation will be ''G''-integral.<ref>{{harvnb|Hall|2015}} Proposition 12.5</ref> For ''G'' semisimple, the set of all ''G''-integral weights is a sublattice ''P''(''G'') ⊂ ''P''(<math>\mathfrak g</math>). If ''G'' is simply connected, then ''P''(''G'') = ''P''(<math>\mathfrak g</math>). If ''G'' is not simply connected, then the lattice ''P''(''G'') is smaller than ''P''(<math>\mathfrak g</math>) and their quotient is isomorphic to the [[Fundamental group|fundamental group]] of ''G''.<ref>{{harvnb|Hall|2015}} Corollary 13.8 and Corollary 13.20</ref>

===Partial ordering on the space of weights===
[[File:Illustration_of_notion_of_"higher"_for_root_systems.png|thumb|right|If the positive roots are <math>\alpha_1</math>, <math>\alpha_2</math>, and <math>\alpha_3</math>, the shaded region is the set of points higher than <math>\lambda</math>]]
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 <math>\mathfrak g</math>. Recall that ''R'' is the set of roots; we now fix a set <math>R^+</math> of [[Root system#Positive roots and simple roots|positive roots]].

Consider two elements <math>\mu</math> and <math>\lambda</math> of  <math>\mathfrak h_0</math>. We are mainly interested in the case where <math>\mu</math> and <math>\lambda</math> are integral, but this assumption is not necessary to the definition we are about to introduce. We then say that <math>\mu</math> is '''higher''' than <math>\lambda</math>, which we write as <math>\mu\succeq\lambda</math>, if <math>\mu-\lambda</math> is expressible as a linear combination of positive roots with non-negative real coefficients.<ref>{{harvnb|Hall|2015}} Definition 8.39</ref> This means, roughly, that "higher" means in the directions of the positive roots. We equivalently say that <math>\lambda</math> is "lower" than <math>\mu</math>, which we write as <math>\lambda\preceq\mu</math>.

This is only a ''partial'' ordering; it can easily happen that <math>\mu</math> is neither higher nor lower than <math>\lambda</math>.

===Dominant weight===
An integral element λ is ''dominant'' if <math>\langle\lambda,\gamma\rangle\geq 0</math> for each positive root ''γ''. Equivalently, λ is dominant if it is a ''non-negative'' integer combination of the fundamental weights. In the <math>A_2</math> 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 <math>\langle\lambda,\gamma\rangle\geq 0</math> is known as the ''fundamental Weyl chamber'' associated to the given set of positive roots.

===Theorem of the highest weight===
A weight <math>\lambda</math> of a representation <math>V</math> of <math>\mathfrak g</math> is called a '''highest weight''' if every other weight of <math>V</math> is lower than <math>\lambda</math>.

The theory [[Lie algebra representation#Classifying finite-dimensional representations of Lie algebras|classifying the finite-dimensional irreducible representations]] of <math>\mathfrak g</math> is by means of a "theorem of the highest weight." The theorem says that<ref>{{harvnb|Hall|2015}} Theorems 9.4 and 9.5</ref>
:(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 module]]s.

===Highest-weight module===
A representation (not necessarily finite dimensional) ''V'' of <math>\mathfrak g</math> 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 <math>\mathfrak g</math>. Every irreducible <math>\mathfrak g</math>-module with a highest weight is necessarily a highest-weight module, but in the infinite-dimensional case, a highest weight module need not be irreducible. 
For each <math>\lambda\in\mathfrak h^*</math>—not necessarily dominant or integral—there exists a unique (up to isomorphism) simple highest-weight <math>\mathfrak g</math>-module with highest weight λ, which is denoted ''L''(λ), but this module is infinite dimensional unless λ is dominant integral. It can be shown that each highest weight module with highest weight λ is a [[Quotient module|quotient]] of the [[Verma module]] ''M''(λ). This is just a restatement of ''universality property'' in the definition of a Verma module.

Every ''finite-dimensional'' highest weight module is irreducible.<ref>This follows from (the proof of) Proposition 6.13 in {{harvnb|Hall|2015}} together with the general result on complete reducibility of finite-dimensional representations of semisimple Lie algebras</ref>

==See also==
*[[Lie algebra representation#Classifying finite-dimensional representations of Lie algebras|Classifying finite-dimensional representations of Lie algebras]]
*[[Compact group#Representation theory of a connected compact Lie group|Representation theory of a connected compact Lie group]]
*[[Highest-weight category]]
*[[Root system]]

==Notes==
{{reflist|group=note}}

==References==
{{reflist}}
{{ref begin}}

* {{Fulton-Harris}}.
* {{citation|last2=Wallach|first2=Nolan R.|last1=Goodman|first1=Roe|year=1998|title=Representations and Invariants of the Classical Groups|publisher= Cambridge University Press|isbn= 978-0-521-66348-9}}.
*{{Citation| last=Hall|first=Brian C.|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|edition=2nd|series=Graduate Texts in Mathematics|volume=222|publisher=Springer|year=2015|isbn=978-3319134666}}
* {{citation|title=Introduction to Lie Algebras and Representation Theory|first=James E.|last=Humphreys|publisher=Birkhäuser|year=1972a|isbn=978-0-387-90053-7|url-access=registration|url=https://archive.org/details/introductiontoli00jame}}.
* {{citation | last1=Humphreys | first1=James E. | title=Linear Algebraic Groups | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-90108-4 |mr=0396773 | year=1972b | volume=21}}
* {{citation|title=Lie Groups Beyond an Introduction|first=Anthony W.|last= Knapp|edition=2nd|publisher=Birkhäuser|year= 2002| isbn=978-0-8176-4259-4}}.
<!--* {{citation|last1=Roggenkamp|first1= K.|last2= Stefanescu|first2= M.|title= Algebra &ndash; Representation Theory|publisher= Springer|year= 2002|isbn=978-0-7923-7113-7}}.--><!--Not sure what this is useful for: it is the proceedings of a conference-->

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[[Category:Lie algebras]]
[[Category:Representation theory of Lie algebras]]
[[Category:Representation theory of Lie groups]]

{{Sourceattribution|Weight (representation theory)}}