Algebraic character

From Wikipedia, the free encyclopedia

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.

1 Definition
2 Example
3 Properties
4 Generalization
5 See also
6 References

[edit] 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 $e μ$ , 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 .

[edit] Example

The algebraic character of the Verma module $M λ$ with the highest weight $λ$ 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.

[edit] 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.