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.
be a semisimple Lie algebra with a fixed Cartan subalgebra
consist of the (possibly infinite) formal integral linear combinations of
, the (complex) vector space of weights.
defined by the formula: where the sum is taken over all weight spaces of the module
The algebraic character of the Verma module
is given by the formula with the product taken over the set of positive roots.
On the other hand, although one can define multiplication of the formal exponents by the formula
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.
Characters also can be defined almost verbatim for weight modules over a Kac–Moody or generalized Kac–Moody Lie algebra.