Algebraic character explained
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 (possibly infinite) formal integral linear combinations of
, where
, the (complex) 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:
ch(V)=\sum\mu\dimV\mue\mu,
where the sum is taken over all weight spaces of the module
Example
with the highest weight
is given by the formula
ch(Mλ)=
| eλ |
| \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
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.
See also
References
