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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''.
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| 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>
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| 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'' = 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.)
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| ===Weight space of a representation===
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| 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>
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| 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
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| :<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]].
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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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| 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.
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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 (root system)|simple root]]s.
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| ===Ordering on the space of weights===
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| Let '''h'''*<sub>0</sub> be the real subspace of '''h'''* (if it is complex) generated by the roots of '''g'''.
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| There are two concepts how to define an [[Order theory|ordering]] of '''h'''*<sub>0</sub>.
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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 (root system)|simple root]]s.
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| The second concept is given by an element ''f'' in '''h'''<sub>0</sub> 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 λ(''H''<sub>γ</sub>) ∈ '''Z''' 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]]{{disambiguation needed|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 ''t'' in '''h''' 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 ''P''(''G'') ⊂ ''P''('''g'''). 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===
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| 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===
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| A weight λ of a representation ''V'' is called the ''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 dominant 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--> | |
| {{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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Inside company Center (where you can warm up and enjoy some hot chocolate and refreshments) there will be several hands-on activities and demonstrations for those ages to enjoy. Activities include learning how to make rope, dressing up like soldiers at the kids' clothing station, trying hand sewing and watching period carpenters work. The Museum features many artifacts and interactive displays concerning Ohio's role in the war of 1812.
If you hiking by using your romantic partner, you can save weight by bringing one particular sleeping luggage. This works for warmer night times. Just open up and make use of the bag on the two of you, as a blanket. With decent sleeping pads, by work well even on cooler nights.
I stepped back towards board and started 3rd column of nails, starting on the top of the board and dealing down. I taped the tip of the nail-gun about the flexible wood and let your catch push my hand, and the gun, back once again. With each quick grunt on the nail-gun I marveled at how easy it would join two pieces of wood so solidly. In my small second to last shot in that column, I heard Adam yell through your other side of the wall directly after the nail-gun grunted.
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