Algebraic character

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 formula_1 be a semisimple Lie algebra with a fixed Cartan subalgebra formula_2 and let the abelian group formula_3 consist of the (possibly infinite) formal integral linear combinations of formula_4, where formula_5, the (complex) vector space of weights. Suppose that formula_6 is a locally-finite weight module. Then the algebraic character of formula_6 is an element of formula_8
defined by the formula:
where the sum is taken over all weight spaces of the module formula_10
Example.
The algebraic character of the Verma module formula_11 with the highest weight formula_12 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 formula_14 and extend it to their "finite" linear combinations by linearity, this does not make formula_8 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.