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 ${\displaystyle {\mathfrak {g}}}$ be a semisimple Lie algebra with a fixed Cartan subalgebra ${\displaystyle {\mathfrak {h}},}$ and let the abelian group ${\displaystyle A=\mathbb {Z} [[{\mathfrak {h}}^{*}]]}$ consist of the (possibly infinite) formal integral linear combinations of ${\displaystyle e^{\mu }}$, where ${\displaystyle \mu \in {\mathfrak {h}}^{*}}$, the (complex) vector space of weights. Suppose that ${\displaystyle V}$ is a locally-finite weight module. Then the algebraic character of ${\displaystyle V}$ is an element of ${\displaystyle A}$ defined by the formula:

${\displaystyle ch(V)=\sum _{\mu }\dim V_{\mu }e^{\mu },}$

where the sum is taken over all weight spaces of the module ${\displaystyle V.}$

Example[]

The algebraic character of the Verma module ${\displaystyle M_{\lambda }}$ with the highest weight ${\displaystyle \lambda }$ is given by the formula

${\displaystyle ch(M_{\lambda })={\frac {e^{\lambda }}{\prod _{\alpha >0}(1-e^{-\alpha })}},}$

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 ${\displaystyle e^{\mu }\cdot e^{\nu }=e^{\mu +\nu }}$ and extend it to their finite linear combinations by linearity, this does not make ${\displaystyle A}$ 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.

See also[]

• Algebraic representation
• Weyl-Kac character formula

References[]

• Weyl, Hermann (1946). The Classical Groups: Their Invariants and Representations. Princeton University Press. ISBN 0-691-05756-7. Retrieved 2007-03-26.
• Kac, Victor G (1990). Infinite-Dimensional Lie Algebras. Cambridge University Press. ISBN 0-521-46693-8. Retrieved 2007-03-26.
• Wallach, Nolan R; Goodman, Roe (1998). Representations and Invariants of the Classical Groups. Cambridge University Press. ISBN 0-521-66348-2. Retrieved 2007-03-26.