Algebraic character

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 ${\mathfrak {g}}$ be a semisimple Lie algebra with a fixed Cartan 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 $\mu \in {\mathfrak {h}}^{*}$ , 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:

$ch(V)=\sum _{\mu }\dim V_{\mu }e^{\mu },$ where the sum is taken over all weight spaces 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 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 $e^{\mu }\cdot e^{\nu }=e^{\mu +\nu }$ 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.