Algebraic character

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 be a semisimple Lie algebra with a fixed Cartan subalgebra and let the abelian group consist of the formal integral linear combinations of, where, the vector space of weights. Suppose that is a locally-finite weight module. Then the algebraic character of is an element of
defined by the formula:
where the sum is taken over all Weight [space (representation theory)|weight spaces] of the module

Example

The algebraic character of the Verma module with the highest weight is given by the formula
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 and extend it to their finite linear combinations by linearity, this does not make 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.