1.Three definitions
Let $\mathfrak g$ be a Lie algebra over a field $k$. Let $(V, \rho)$ be a $\mathfrak g$-representation.
In class I was presented with various definitions of characters of $(V, \rho)$.
- First case: $(V, \rho)$ a one-dimensional $\mathfrak g$-representation.
By identifying $\mathfrak {gl}(k) \cong k$ and using that $\rho$ is a Lie algebra morphism one obtains (with the universal property of quotients) a bijection between $(\mathfrak g /[\mathfrak g, \mathfrak g])^*$ and one dimensional $\mathfrak g$-representations. We call the linear form in $(\mathfrak g /[\mathfrak g, \mathfrak g])^*$ corresponding to a $\mathfrak g$-rep $(V, \rho)$ its character. - Second case: $\mathfrak g = \mathfrak sl(2,\mathbb C)$.
Since $\mathfrak g$ is semisimple by Weyl's theorem any $\mathfrak g$-rep is semisimple. The classification of simple $\mathfrak g$-reps gives that hence any $\mathfrak g$-rep decomposes (as a vector space) into integral weight spaces $V_\lambda$ of $\rho(h)$ where $$h = \begin{pmatrix} 1 & 0 \\ 0 &-1 \\ \end{pmatrix} \in \mathfrak g .$$ We call the Laurent polynomial $\sum_{\lambda \in \mathbb Z} dim(V_\lambda) q^i \in \mathbb Z[q,q^{-1}$] the character of the $\mathfrak g$-rep $(V, \rho)$. - Third case: $\mathfrak g$ a complex semisimple Lie algebra & $(V, \rho)$ a finite-dimensional $\mathfrak g$-rep.
Let $\mathfrak h$ be a Cartan subalgebra of $\mathfrak g$. One proves that $V$ is a weight module, i.e. that it has the weight space decomposition $V=\bigoplus _{\lambda \in \mathfrak h^*}V_\lambda$. Consider the group algebra $\mathbb Z[\mathfrak h^*]$ of the group $\mathfrak h^*$ with respect to addition. Its elements are formal $\mathbb Z$-linear combinations of basis elements $e^\lambda$ for $\lambda \in \mathfrak h^*$. (More formally, the group algebra is the $\mathbb Z$-algebra of functions $\mathfrak h^* \rightarrow \mathbb Z$ of finite support with $\mathbb Z$-basis consisting of functions $e^\lambda: \mathfrak h^* \rightarrow \mathbb Z$with $e^\lambda(\lambda)=1$ and $e^\lambda(\mu)=0$ for $\mu \neq \lambda$.) We call $\sum _{\lambda \in \mathfrak h ^*} dim(V_\lambda) e^\lambda \in \mathbb Z[\mathfrak h^*]$ the character of $(V, \rho)$.
2. Question
(How) are these notions related?
