# Characters of Lie algebra representations

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?

• Shouldn't the polynomial in the second case only summing over weights of the finite dimensional representation (instead of summing over all weights/integers)? Commented Mar 23, 2023 at 7:41