Formal characters are additive and multiplicative

Given: The Axiom of Choice, finite-dimensional $\mathfrak{g}$-modules $V,W$, a short exact sequence $0\to V^{\mathrm{\prime }}\to V\to V^{\mathrm{\prime }\mathrm{\prime }}\to 0$ of such modules, and the completing ring $\mathcal{R}$.

The Axiom of Choice is assumed; it enters through the published decomposition and category suppliers cited below (The Axiom of Choice).

$\mathrm{ch}U={\sum }_{\mu }(\dim U_{\mu })e^{\mu }$ is a finite-support element of $\mathcal{R}$ for every finite-dimensional $\mathfrak{g}$-module $U$ (The formal character of a finite-dimensional weight module).

Taking weight spaces is exact on $\mathfrak{h}$-semisimple modules: a $\mathfrak{g}$-linear map preserves weight spaces, so for every $\mu$ the sequence $0\to V_{\mu }^{\mathrm{\prime }}\to V_{\mu }\to V_{\mu }^{\mathrm{\prime }\mathrm{\prime }}\to 0$ is exact, and every $\mathfrak{g}$-module in sight has a weight-space decomposition; $V$, $V^{\mathrm{\prime }}$, $V^{\mathrm{\prime }\mathrm{\prime }}$ and $V\oplus W$ are objects of the category $\mathcal{O}$, and finite-dimensional $\mathfrak{h}$-semisimple modules belong to $\mathcal{O}$ (The Grothendieck group and character of O, Verma and finite-dimensional weight modules belong to O, Finite-dimensional tensoring preserves O).

For the diagonal action on $V\otimes W$ of Direct-sum, dual, Hom, and tensor representations and $H\in \mathfrak{h}$ one has $H\cdot (v\otimes w)=(Hv)\otimes w+v\otimes (Hw)$, so $V_{\mu }\otimes W_{\nu }\subseteq (V\otimes W{)}_{\mu +\nu }$ and, choosing bases of weight vectors in $V$ and in $W$, the weight spaces of $V\otimes W$ are $(V\otimes W{)}_{\eta }={⨁}_{\mu +\nu =\eta }{V}_{\mu }\otimes W_{\nu }$ (Weight and weight space, Direct-sum, dual, Hom, and tensor representations).

The product in $\mathcal{R}$ is the convolution $(fg{)}_{\eta }={\sum }_{\mu +\nu =\eta }{c}_{\mu }{d}_{\nu }$ with $f={\sum }_{\mu }{c}_{\mu }{e}^{\mu }$, $g={\sum }_{\nu }{d}_{\nu }{e}^{\nu }$ (The completed formal character ring).