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mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, a weight of an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
''A'' over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
F is an
algebra homomorphism In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in , * F(kx) = kF ...
from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a
multiplicative character In mathematics, a multiplicative character (or linear character, or simply character) on a group ''G'' is a group homomorphism from ''G'' to the multiplicative group of a field , usually the field of complex numbers. If ''G'' is any group, then the ...
of a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
. The importance of the concept, however, stems from its application to
representations ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
of
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s and hence also to
representations ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
of algebraic and
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s. In this context, a weight of a representation is a generalization of the notion of an
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
, and the corresponding
eigenspace In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
is called a weight space.


Motivation and general concept

Given a set ''S'' of n\times n
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
over the same field, each of which is
diagonalizable In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
, and any two of which
commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
, it is always possible to simultaneously diagonalize all of the elements of ''S''.In fact, given a set of commuting matrices over an algebraically closed field, they are
simultaneously triangularizable In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are ...
, without needing to assume that they are diagonalizable.
Equivalently, for any set ''S'' of mutually commuting
semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s of a finite-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
''V'' there exists a basis of ''V'' consisting of ''simultaneous
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s'' of all elements of ''S''. Each of these common eigenvectors ''v'' ∈ ''V'' defines a
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the s ...
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 mathematics, a multiplicative character (or linear character, or simply character) on a group ''G'' is a group homomorphism from ''G'' to the multiplicative group of a field , usually the field of complex numbers. If ''G'' is any group, then the ...
in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, which is a homomorphism ''χ'' from a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
''G'' to the
multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred to ...
of a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
F. Thus ''χ'': ''G'' → F× satisfies ''χ''(''e'') = 1 (where ''e'' is the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
of ''G'') and : \chi(gh) = \chi(g)\chi(h) for all ''g'', ''h'' in ''G''. Indeed, if ''G''
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
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 Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
''A'' over F, by replacing ''χ'': ''G'' → F× by a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
''χ'': ''A'' → F with: : \chi(ab) = \chi(a)\chi(b) for all ''a'', ''b'' in ''A''. If an algebra ''A''
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
on a vector space ''V'' over F to any simultaneous eigenspace, this corresponds an
algebra homomorphism In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in , * F(kx) = kF ...
from ''A'' to F assigning to each element of ''A'' its eigenvalue. If ''A'' is a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
(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 In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
; but since F is commutative this simply means that this map must vanish on Lie brackets: ''χ''( ,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 In mathematics, a Lie algebra \mathfrak is solvable if its derived series terminates in the zero subalgebra. The ''derived Lie algebra'' of the Lie algebra \mathfrak is the subalgebra of \mathfrak, denoted : mathfrak,\mathfrak/math> that consi ...
''g,gand hence descends to a weight on the
abelian Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
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 In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
or an
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
, 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 \mathfrak g be a complex semisimple Lie algebra and \mathfrak h a Cartan subalgebra of \mathfrak g. In this section, we describe the concepts needed to formulate the "theorem of the highest weight" classifying the finite-dimensional representations of \mathfrak g. 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 ''V'' be a representation of a Lie algebra \mathfrak g over C and let λ be a linear functional on \mathfrak h. Then the of ''V'' with weight λ is the subspace V_\lambda given by :V_\lambda:=\. A weight of the representation ''V'' 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 \mathfrak h, with the corresponding eigenvalues given by λ. If ''V'' is the direct sum of its weight spaces :V=\bigoplus_ V_\lambda then it is called a '';'' this corresponds to there being a common
eigenbasis In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
(a basis of simultaneous eigenvectors) for all the represented elements of the algebra, i.e., to there being simultaneously diagonalizable matrices (see
diagonalizable matrix In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) ...
). If ''G'' is group with Lie algebra \mathfrak g, every finite-dimensional representation of ''G'' induces a representation of \mathfrak g. A weight of the representation of ''G'' is then simply a weight of the associated representation of \mathfrak g. 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

If ''V'' is the
adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
of \mathfrak g, the nonzero weights of ''V'' are called
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
, the weight spaces are called root spaces, and weight vectors are called root vectors. Explicitly, a linear functional \alpha on \mathfrak h is called a root if \alpha\neq 0 and there exists a nonzero X in \mathfrak g such that : ,X\alpha(H)X for all H in \mathfrak h. The collection of roots forms a
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representati ...
. From the perspective of representation theory, the significance of the roots and root vectors is the following elementary but important result: If ''V'' is a representation of \mathfrak g, ''v'' is a weight vector with weight \lambda and ''X'' is a root vector with root \alpha, then :H\cdot (X\cdot v)= \lambda+\alpha)(H)X\cdot v) for all ''H'' in \mathfrak h. That is, X\cdot v is either the zero vector or a weight vector with weight \lambda+\alpha. Thus, the action of X maps the weight space with weight \lambda into the weight space with weight \lambda+\alpha.


Integral element

Let \mathfrak h^*_0 be the real subspace of \mathfrak h^* generated by the roots of \mathfrak g. 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 \mathfrak h^*_0 with a subspace \mathfrak h_0 of \mathfrak h. With this identification, the coroot associated to a root \alpha is given as :H_\alpha=2\frac. We now define two different notions of integrality for elements of \mathfrak h_0. The motivation for these definitions is simple: The weights of finite-dimensional representations of \mathfrak g satisfy the first integrality condition, while if ''G'' is a group with Lie algebra \mathfrak g, the weights of finite-dimensional representations of ''G'' satisfy the second integrality condition. An element \lambda\in\mathfrak h_0 is algebraically integral if :\langle\lambda,H_\alpha\rangle=2\frac\in\mathbf for all roots \alpha. The motivation for this condition is that the coroot H_\alpha can be identified with the ''H'' element in a standard basis for an sl(2,C)-subalgebra of g. By elementary results for sl(2,C), the eigenvalues of H_\alpha in any finite-dimensional representation must be an integer. We conclude that, as stated above, the weight of any finite-dimensional representation of \mathfrak g is algebraically integral. The fundamental weights \omega_1,\ldots,\omega_n are defined by the property that they form a basis of \mathfrak h_0 dual to the set of coroots associated to the simple roots. That is, the fundamental weights are defined by the condition :2\frac=\delta_ where \alpha_1,\ldots\alpha_n are the simple roots. An element \lambda is then algebraically integral if and only if it is an integral combination of the fundamental weights. The set of all \mathfrak g-integral weights is a
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
in \mathfrak h_0 called the ''weight lattice'' for \mathfrak g, denoted by P(\mathfrak g). The figure shows the example of the Lie algebra sl(3,C), whose root system is the A_2 root system. There are two simple roots, \gamma_1 and \gamma_2. The first fundamental weight, \omega_1, should be orthogonal to \gamma_2 and should project orthogonally to half of \gamma_1, and similarly for \omega_2. The weight lattice is then the triangular lattice. Suppose now that the Lie algebra \mathfrak g is the Lie algebra of a Lie group ''G''. Then we say that \lambda\in\mathfrak h_0 is analytically integral (''G-integral'') if for each ''t'' in \mathfrak h such that \exp(t)=1\in G we have \langle\lambda,t\rangle\in 2\pi i \mathbf. The reason for making this definition is that if a representation of \mathfrak g arises from a representation of ''G'', then the weights of the representation will be ''G''-integral. For ''G'' semisimple, the set of all ''G''-integral weights is a sublattice ''P''(''G'') ⊂ ''P''(\mathfrak g). If ''G'' is
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
, 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 In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
is isomorphic to the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of ''G''.


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 g. Recall that ''R'' is the set of roots; we now fix a set R^+ of
positive roots In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation ...
. Consider two elements \mu and \lambda of \mathfrak h_0. We are mainly interested in the case where \mu and \lambda are integral, but this assumption is not necessary to the definition we are about to introduce. We then say that \mu is higher than \lambda, which we write as \mu\succeq\lambda, if \mu-\lambda is expressible as a linear combination of positive roots with non-negative real coefficients. This means, roughly, that "higher" means in the directions of the positive roots. We equivalently say that \lambda is "lower" than \mu, which we write as \lambda\preceq\mu. This is only a ''partial'' ordering; it can easily happen that \mu is neither higher nor lower than \lambda.


Dominant weight

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


Theorem of the highest weight

A weight \lambda of a representation V of \mathfrak g is called a highest weight if every other weight of V is lower than \lambda. The theory classifying the finite-dimensional irreducible representations of \mathfrak g is by means of a "theorem of the highest weight." The theorem says that :(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 Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics. Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Spe ...
s.


Highest-weight module

A representation (not necessarily finite dimensional) ''V'' of \mathfrak 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 In mathematics, a root system is a configuration of vector space, vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and ...
spaces in \mathfrak g. Every irreducible \mathfrak g-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 \lambda\in\mathfrak h^*—not necessarily dominant or integral—there exists a unique (up to isomorphism)
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
highest-weight \mathfrak g-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 In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of the
Verma module Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics. Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Spe ...
''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.This follows from (the proof of) Proposition 6.13 in together with the general result on complete reducibility of finite-dimensional representations of semisimple Lie algebras


See also

* Classifying finite-dimensional representations of Lie algebras * Representation theory of a connected compact Lie group * Highest-weight category *
Root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representati ...


Notes


References

* . * . * * . * * . {{refend Lie algebras Representation theory of Lie algebras Representation theory of Lie groups