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| '''Algebraic character''' is a formal expression attached to a module in [[representation theory]] of [[semisimple Lie algebra]]s that generalizes the [[Weyl character formula|character of a finite-dimensional representation]] and is analogous to the [[Harish-Chandra character]] of the representations of [[semisimple Lie group]]s.
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| == Definition ==
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| Let <math>\mathfrak{g}</math> be a [[semisimple Lie algebra]] with a fixed [[Cartan subalgebra]] <math>\mathfrak{h},</math> and let the abelian group <math>A=\mathbb{Z}[[\mathfrak{h}^*]]</math> consist of the (possibly infinite) formal integral linear combinations of <math>e^{\mu}</math>, where <math>\mu\in\mathfrak{h}^*</math>, the (complex) vector space of weights. Suppose that <math>V</math> is a locally-finite [[weight module]]. Then the algebraic character of <math>V</math> is an element of <math>A</math>
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| defined by the formula:
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| : <math> ch(V)=\sum_{\mu}\dim V_{\mu}e^{\mu}, </math>
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| where the sum is taken over all [[weight space]]s of the module <math>V.</math>
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| == Example ==
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| The algebraic character of the [[Verma module]] <math>M_\lambda</math> with the highest weight <math>\lambda</math> is given by the formula
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| : <math> ch(M_{\lambda})=\frac{e^{\lambda}}{\prod_{\alpha>0}(1-e^{-\alpha})},</math>
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| with the product taken over the set of positive roots.
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| == Properties ==
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| Algebraic characters are defined for locally-finite [[weight module]]s and are ''additive'', i.e. the character of a direct sum of modules is the sum of their characters. On the other hand, although one can define multiplication of the formal exponents by the formula <math>e^{\mu}\cdot e^{\nu}=e^{\mu+\nu}</math> and extend it to their ''finite'' linear combinations by linearity, this does not make <math>A</math> into a ring, because of the possibility of formal infinite sums. Thus the product of algebraic characters is well defined only in restricted situations; for example, for the case of a [[highest weight module]], or a finite-dimensional module. In good situations, the algebraic character is ''multiplicative'', i.e., the character of the tensor product of two weight modules is the product of their characters.
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| == Generalization ==
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| Characters also can be defined almost ''verbatim'' for weight modules over a [[Kac-Moody algebra|Kac-Moody]] or [[generalized Kac-Moody algebra|generalized Kac-Moody]] Lie algebra.
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| == See also ==
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| *[[Weyl character formula#Weyl–Kac character formula|Weyl-Kac character formula]]
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| ==References==
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| *{{cite book|last = Weyl|first = Hermann|title = The Classical Groups: Their Invariants and Representations|publisher = Princeton University Press|year = 1946|isbn = 0-691-05756-7|url = http://books.google.com/books?id=zmzKSP2xTtYC|accessdate = 2007-03-26}}
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| *{{cite book|last = Kac|first = Victor G|title = Infinite-Dimensional Lie Algebras|publisher = Cambridge University Press|year = 1990|isbn = 0-521-46693-8|url = http://books.google.com/books?id=kuEjSb9teJwC|accessdate = 2007-03-26}}
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| *{{cite book|last = Wallach|first = Nolan R|coauthors = Goodman, Roe|title = Representations and Invariants of the Classical Groups|publisher = Cambridge University Press|year = 1998|isbn = 0-521-66348-2|url = http://books.google.com/books?vid=ISBN0521663482&id=MYFepb2yq1wC|accessdate = 2007-03-26}}
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| [[Category:Lie algebras]]
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| [[Category:Representation theory of Lie algebras]]
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