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| In the [[mathematics|mathematical]] field of [[representation theory]], a '''weight''' of an [[algebra over a field|algebra]] ''A'' over a field '''F''' is an [[algebra homomorphism]] from ''A'' to '''F''' – a [[linear functional]] – or equivalently, a one dimensional [[Representation (mathematics)|representation]] of ''A'' over '''F'''. It is the algebra analogue of a [[multiplicative character]] of a [[group (mathematics)|group]]. The importance of the concept, however, stems from its application to [[Lie algebra representation|representations]] of [[Lie algebra]]s 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'''.
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| ==Motivation and general concept==
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| ===Weights===
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| Given a set ''S'' of [[matrix (mathematics)|matrices]], 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">The converse is also true – a set of diagonalizable matrices commutes if and only if the set is simultaneously diagonalisable {{harv|Horn|Johnson|1985|pp=51–53}}.</ref><ref group="note">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.</ref> Equivalently, for any set ''S'' of mutually commuting [[semisimple operator|semisimple]] [[linear transformation]]s of a finite-dimensional [[vector space]] ''V'' there exists a basis of ''V'' consisting of ''{{anchor|simultaneous eigenvector}}simultaneous [[eigenvector]]s'' 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.
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| The notion is closely related to the idea of a [[multiplicative character]] in [[group theory]], which is a homomorphism ''χ'' from a [[group (mathematics)|group]] ''G'' to the [[multiplicative group]] of a [[field (mathematics)|field]] '''F'''. Thus ''χ'': ''G'' → '''F'''<sup>×</sup> satisfies ''χ''(''e'') = 1 (where ''e'' is the [[identity element]] of ''G'') and
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| :<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.
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| The notion of multiplicative character can be extended to any [[algebra over a field|algebra]] ''A'' over '''F''', by replacing ''χ'': ''G'' → '''F'''<sup>×</sup> by a [[linear map]] ''χ'': ''A'' → '''F''' with:
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| :<math> \chi(ab) = \chi(a)\chi(b)</math> | | :<math forcemathmode="mathml">E=mc^2</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 corresponds an [[algebra homomorphism]] from ''A'' to '''F''' assigning to each element of ''A'' its eigenvalue.
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| 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 [[commutator]]s : ''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.
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| If ''G'' is a [[Lie group]] or an [[algebraic group]], then a multiplicative character ''θ'': ''G'' → '''F'''<sup>×</sup> induces a weight ''χ'' = d''θ'': '''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.)
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| | :<math forcemathmode="source">E=mc^2</math> --> |
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| ===Weight space of a representation=== | | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
| 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
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| :<math>V_\lambda:=\{v\in V: \forall \xi\in \mathfrak{g},\quad \xi\cdot v=\lambda(\xi)v\}.</math> | |
| 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''.
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| If ''V'' is the direct sum of its weight spaces
| | ==Demos== |
| :<math>V=\bigoplus_{\lambda\in\mathfrak{g}^*} V_\lambda</math>
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| then it is called a ''{{visible anchor|weight module}};'' this corresponds to having an [[eigenbasis]] (a basis of eigenvectors), i.e., being a [[diagonalizable matrix]].
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| Similarly, we can define a weight space ''V''<sub>''λ''</sub> for any [[representation theory|representation]] of a [[Lie group]] or an [[associative algebra]].
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
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| ==Semisimple Lie algebras==
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| Let '''g''' be a [[Lie algebra]], '''h''' a [[maximal]] [[commutative]] [[Lie subalgebra]] consisting of [[Semisimple Lie algebra|semi-simple]] elements (sometimes called [[Cartan subalgebra]]) and let ''V'' be a finite dimensional representation of '''g'''. If '''g''' is [[semisimple Lie algebra|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'''.
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| 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''.
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| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| If ''V'' is the [[Adjoint representation of a Lie algebra|adjoint representation]] of '''g''', its weights are called [[root system|roots]], the weight spaces are called root spaces, and weight vectors are sometimes called root vectors.
| | ==Test pages == |
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| 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 root]]s Φ<sup>+</sup> has been fixed. This is equivalent to the choice of a set of [[simple root]]s.
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| ===Ordering on the space of weights===
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| Let <math>\mathfrak{h}_0^*</math> be the real subspace of '''h'''* (if it is complex) generated by the roots of '''g'''.
| | *[[Url2Image|Url2Image (private Wikis only)]] |
| | | ==Bug reporting== |
| There are two concepts how to define an [[Order theory|ordering]] of <math>\mathfrak{h}_0^*</math>.
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| The first one is
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| :''μ'' ≤ ''λ'' if and only if ''λ'' − ''μ'' is nonnegative linear combination of [[simple root]]s.
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| The second concept is given by an element <math>f\in\mathfrak{h}_0</math> and
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| :''μ'' ≤ ''λ'' if and only if ''μ''(''f'') ≤ ''λ''(''f'').
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| Usually, ''f'' is chosen so that ''β''(''f'') > 0 for each [[positive root]] ''β''.
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| ===Integral weight===
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| A weight ''λ'' ∈ '''h'''* is ''integral'' (or '''g'''-integral), if <math>\lambda(H_\gamma)\in\Z</math> for each [[coroot]] ''H''<sub>γ</sub> such that γ is a positive root.
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| The fundamental weights <math>\omega_1,\ldots,\omega_n</math> are defined by the property that they form a basis of '''h'''* dual to the set of [[simple coroot]]s <math>H_{\alpha_1}, \ldots, H_{\alpha_n}</math>.
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| Hence ''λ'' is integral if it is an integral combination of the fundamental weights. The set of all '''g'''-integral weights is a [[lattice (mathematics)|lattice]]{{dn|date=May 2012}} in '''h'''* called ''weight lattice'' for '''g''', denoted by ''P''('''g''').
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| A weight ''λ'' of the Lie group ''G'' is called integral, if for each <math>t\in\mathfrak{h}</math> such that <math>\exp(t)=1\in G,\,\,\lambda(t)\in 2\pi i \mathbf{Z}</math>. For ''G'' semisimple, the set of all ''G''-integral weights is a sublattice <math>P(G)\subset P(\mathfrak{g})</math>. 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 (group theory)|quotient]] is isomorphic to the [[fundamental group]] of ''G''.
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| ===Dominant weight=== | |
| A weight ''λ'' is ''dominant'' if <math>\lambda(H_\gamma)\geq 0</math> for each [[coroot]] ''H''<sub>γ</sub> such that ''γ'' is a positive root. Equivalently, ''λ'' is dominant, if it is a non-negative linear combination of the [[weight (representation theory)#fundamental weight|fundamental weight]]s.
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| The [[convex hull]] of the dominant weights is sometimes called the ''fundamental Weyl chamber''.
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| Sometimes, the term ''dominant weight'' is used to denote a dominant (in the above sense) and [[integral weight]].
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| ===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".
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| Similarly, we define the ''lowest weight''.
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| 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.
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| Then, a representation is said to have ''highest weight'' ''λ'' if ''λ'' is a weight and all its other weights are less than ''λ''.
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| Similarly, it is said to have ''lowest weight'' ''λ'' if ''λ'' is a weight and all its other weights are greater than it.
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| A weight vector <math>v_\lambda \in V</math> of weight ''λ'' is called a ''highest-weight vector'', or ''vector of highest weight'', if all other weights of ''V'' are smaller than ''λ''.
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| ===Highest-weight module===
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| 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'''.
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| This is something more special than a '''g'''-[[module (mathematics)|module]] with a [[highest weight]].
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| Similarly we can define a highest-weight module for representation of a [[Lie group]] or an [[associative algebra]].
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| ===Verma module===
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| {{main|Verma module}}
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| For each weight ''λ'' ∈ '''h'''*, there exists a unique (up to isomorphism) [[irreducible (representation theory)|simple]] highest-weight '''g'''-module with highest weight ''λ'', which is denoted ''L''(''λ'').
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| 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.
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| A highest-weight module is a weight module. The weight spaces in a highest-weight module are always finite dimensional.
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| ==See also==
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| *[[Highest-weight category]]
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| ==Notes==
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| {{reflist|group=note}}
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| * {{Fulton-Harris}}.
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| * {{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}}.
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| * {{citation | title =Introduction to Lie Algebras and Representation Theory|first=James E.|last=Humphreys|publisher=Birkhäuser|year= 1972a|isbn=978-0-387-90053-7}}.
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| * {{Citation | last1=Horn | first1=Roger A. | last2=Johnson | first2=Charles R. | title=Matrix Analysis | publisher= Cambridge University Press | isbn=978-0-521-38632-6 | year=1985}}<!-- reference for simultaneous diagonalizable -->
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| * {{citation | last1=Humphreys | first1=James E. | title=Linear Algebraic Groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-90108-4 | id={{MathSciNet|id=0396773}} | year=1972b | volume=21}}
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| * {{citation|title=Lie Groups Beyond an Introduction|first=Anthony W.|last= Knapp|edition=2nd|publisher=Birkhäuser|year= 2002| isbn=978-0-8176-4259-4}}.
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| <!--* {{citation|last1=Roggenkamp|first1= K.|last2= Stefanescu|first2= M.|title= Algebra – 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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| {{refend}}
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| [[Category:Lie algebras]]
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| [[Category:Representation theory of Lie algebras]]
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| [[Category:Representation theory of Lie groups]]
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