TITLE: Is there some kind of character theory for representations of finite dimensional algebras?
QUESTION [5 upvotes]: We know that for a representation $V$ of a Lie algebra or a quantum group, we can define character of $V$ as $ch(V)=\sum_{\mu} dim(V_{\mu})e^{\mu}$, where $V_{\mu}$ is the weight space of $V$ with weight $\mu$. I didn't find the corresponding definition for representations of finite dimensional algebras (for example, associative algebras). Is there some kind of character theory for representations of finite dimensional algebras? Thank you very much.

REPLY [7 votes]: Sure. If $A$ is a finite-dimensional algebra over a field $k$ and $V$ a finite-dimensional $A$-module (so we have a morphism $\rho : A \to \text{End}_k(V)$), we can define a character $\chi_V(a) = \text{tr}(\rho(a)) : A \to k$ just as in the case of finite groups (where $A = k[G]$). Characters are additive in short exact sequences, so if $\chi_V$ has a composition series consisting of $n_i$ copies of the simple $A$-module $V_i$, then
$$\chi_V = \sum n_i \chi_{V_i}.$$
Moreover, if $k$ has characteristic zero, the characters corresponding to distinct simple modules are linearly independent. (I don't know a short way to prove this although I am sure there is one; it follows from Artin-Wedderburn applied to $A/J(A)$.) So the character of a module tells you precisely what the simple modules in its composition series are. 
Note that any character $\chi_V$ above is among other things a linear functional $A/[A, A] \to k$ (where $[A, A]$ is the subspace, not the ideal, generated by commutators), so $\dim A/[A, A]$ is an upper bound on the number of simple modules. I don't know if this upper bound is attained in general.