Lie-algebra centers via de-categorification

Let $\mathfrak{g}$ be a Lie algebra over an algebraically closed field $\mathbb{k}$ of characteristic zero. Define the universal grading group $\mathcal{C}(\mathfrak{g})$ as having one generator $g_{\rho }$ for each irreducible $\mathfrak{g}$-representation $\rho$, one relation $g_{\pi }=g_{\rho }^{-1}$ whenever $\pi$ is weakly contained in the dual representation ${\rho }^{\ast }$ (i.e. the kernel of $\pi$ in the enveloping algebra $U(\mathfrak{g})$ contains that of ${\rho }^{\ast }$), and one relation $g_{\rho }=g_{{\rho }^{\mathrm{\prime }}}{g}_{\rho \mathrm{"}}$ whenever $\rho$ is weakly contained in ${\rho }^{\mathrm{\prime }}\otimes \rho \mathrm{"}$. The main result is that attaching to an irreducible representation its central character gives an isomorphism between $\mathcal{C}(\mathfrak{g})$ and the dual ${\mathfrak{z}}^{\ast }$ of the center $\mathfrak{z}\le \mathfrak{g}$ when $\mathfrak{g}$ is (a) finite-dimensional solvable; (b) finite-dimensional semisimple. The group $\mathcal{C}(\mathfrak{g})$ is also trivial when the enveloping algebra $U(\mathfrak{g})$ has a faithful irreducible representation (which happens for instance for various infinite-dimensional algebras of interest, such as $\mathfrak{s}\mathfrak{l}(\mathrm{\infty })$, $\mathfrak{o}(\mathrm{\infty })$ and $\mathfrak{s}\mathfrak{p}(\mathrm{\infty })$). These are analogues of a result of Müger’s for compact groups and a number of results by the author on locally compact groups, and provide further evidence for the pervasiveness of such center-reconstruction phenomena.