- Algebraic character
**Algebraic character**is a formal expression attached to a module inrepresentation theory ofsemisimple Lie algebra s that generalizes the character of a finite-dimensional representation and is analogous to theHarish-Chandra character of the representations ofsemisimple Lie group s.**Definition**Let $mathfrak\{g\}$ be a

semisimple Lie algebra with a fixedCartan subalgebra $mathfrak\{h\},$ and let the abelian group $A=mathbb\{Z\}mathfrak\{h\}^*$ consist of the (possibly infinite) formal integral linear combinations of $e^\{mu\}$, where $muinmathfrak\{h\}^*$, the (complex) vector space of weights. Suppose that $V$ is a locally-finiteweight module . Then the algebraic character of $V$ is an element of $A$defined by the formula:: $ch(V)=sum\_\{mu\}dim\; V\_\{mu\}e^\{mu\},$ where the sum is taken over allweight space s of the module $V.$**Example**The algebraic character of the

Verma module $M\_lambda$ with the highest weight $lambda$ is given by the formula: $ch(M\_\{lambda\})=frac\{e^\{lambda\{prod\_\{alpha>0\}(1-e^\{-alpha\})\},$

with the product taken over the set of positive roots.

**Properties**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 $e^\{mu\}cdot\; e^\{\; u\}=e^\{mu+\; u\}$ and extend it to their "finite" linear combinations by linearity, this does not make $A$ 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 ahighest 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***Weyl-Kac character formula

**References***cite book|last = Weyl|first = Hermann|title = The Classical Groups: Their Invariants and Representations|publisher = Princeton University Press|date = 1946|isbn = 0691057567|url = http://books.google.com/books?id=zmzKSP2xTtYC|accessdate = 2007-03-26

*cite book|last = Kac|first = Victor G|title = Infinite-Dimensional Lie Algebras|publisher = Cambridge University Press|date = 1990|isbn = 0521466938|url = http://books.google.com/books?id=kuEjSb9teJwC|accessdate = 2007-03-26

*cite book|last = Wallach|first = Nolan R|coauthors = Goodman, Roe|title = Representations and Invariants of the Classical Groups|publisher = Cambridge University Press|date = 1998|isbn = 0521663482|url = http://books.google.com/books?vid=ISBN0521663482&id=MYFepb2yq1wC|accessdate = 2007-03-26

*Wikimedia Foundation.
2010.*