# Algebraic character

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{{Short description|Mathematical concept}}

In mathematics, an '''algebraic character''' is a formal expression attached to a module in [representation theory](/source/representation_theory) of [semisimple Lie algebra](/source/semisimple_Lie_algebra)s that generalizes the [character of a finite-dimensional representation](/source/Weyl_character_formula) and is analogous to the [Harish-Chandra character](/source/Harish-Chandra_character) of the representations of [semisimple Lie group](/source/semisimple_Lie_group)s.

== Definition ==
Let <math>\mathfrak{g}</math> be a [semisimple Lie algebra](/source/semisimple_Lie_algebra) with a fixed [Cartan subalgebra](/source/Cartan_subalgebra) <math>\mathfrak{h},</math> and let the abelian group <math>A=\mathbb{Z}[\mathfrak{h}^*](/source/%5Cmathfrak%7Bh%7D%5E*)</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](/source/weight_module). Then the algebraic character of <math>V</math> is an element of <math>A</math>
defined by the formula:
: <math> ch(V)=\sum_{\mu}\dim V_{\mu}e^{\mu}, </math>
where the sum is taken over all [weight spaces](/source/Weight_space_(representation_theory)) of the module <math>V.</math>

== Example ==
The algebraic character of the [Verma module](/source/Verma_module) <math>M_\lambda</math> with the highest weight <math>\lambda</math> is given by the formula

: <math> ch(M_{\lambda})=\frac{e^{\lambda}}{\prod_{\alpha>0}(1-e^{-\alpha})},</math>

with the product taken over the set of positive roots.

== Properties ==
Algebraic characters are defined for locally-finite [weight module](/source/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](/source/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.

== Generalization ==
Characters also can be defined almost ''verbatim'' for weight modules over a [Kac–Moody](/source/Kac%E2%80%93Moody_algebra) or [generalized Kac–Moody](/source/generalized_Kac%E2%80%93Moody_algebra) Lie algebra.

== See also ==
*[Algebraic representation](/source/Algebraic_representation)
*[Weyl-Kac character formula](/source/Weyl_character_formula)

==References==
*{{cite book|last = Weyl|first = Hermann|title = The Classical Groups: Their Invariants and Representations|publisher = Princeton University Press|year = 1953|edition=2nd|isbn = ((0-691-05756-7))<!-- isbn ok, goes to later reprint of same edition by same publisher -->}}
*{{cite book|last = Kac|first = Victor G|title = Infinite-Dimensional Lie Algebras|publisher = Cambridge University Press|year = 1990|isbn = 0-521-46693-8|url = https://books.google.com/books?id=kuEjSb9teJwC|accessdate = 2007-03-26}}
*{{cite book|last = Wallach|first = Nolan R|author2=Goodman, Roe|title = Representations and Invariants of the Classical Groups|publisher = Cambridge University Press|year = 1998|isbn = 0-521-66348-2|url = https://books.google.com/books?id=MYFepb2yq1wC|accessdate = 2007-03-26}}

Category:Lie algebras
Category:Representation theory of Lie algebras

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Adapted from the Wikipedia article [Algebraic character](https://en.wikipedia.org/wiki/Algebraic_character) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Algebraic_character?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.