An algebraic character is a formal expression attached to a module in

representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...

s that generalizes the character of a finite-dimensional representation and is analogous to the Harish-Chandra character of the representations of

semisimple Lie group In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...

Definition

Let

\mathfrak

be a

with a fixed

Cartan subalgebra In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by � ...

\mathfrak,

and let the abelian group

A=\mathbb \mathfrak^*

consist of the (possibly infinite) formal integral linear combinations of

e^

, where

\mu\in\mathfrak^*

, the (complex) vector space of weights. Suppose that

V

is a locally-finite weight module. Then the algebraic character of

V

is an element of

A

defined by the formula: :

ch(V)=\sum_\dim V_e^,

where the sum is taken over all weight spaces of the module

V.

Example

The algebraic character of the

Verma module Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics. Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Sp ...

M_\lambda

with the highest weight

\lambda

is given by the formula :

ch(M_)=\frac,

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

e^\cdot e^=e^

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 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.

References

* * *{{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 Lie algebras Representation theory of Lie algebras

Definition

Example

Properties

Generalization

See also

References