{{Short description|Mathematical concept}}
In mathematics, an '''algebraic character''' is a formal expression attached to a module in representation theory of semisimple Lie algebras that generalizes the character of a finite-dimensional representation and is analogous to the Harish-Chandra character of the representations of semisimple Lie groups.
== Definition == 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> defined by the formula: : <math> ch(V)=\sum_{\mu}\dim V_{\mu}e^{\mu}, </math> where the sum is taken over all weight spaces of the module <math>V.</math>
== Example == The algebraic character of the 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 modules 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.
== Generalization == Characters also can be defined almost ''verbatim'' for weight modules over a Kac–Moody or generalized Kac–Moody Lie algebra.
== See also == *Algebraic representation *Weyl-Kac 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