Lie-algebra centers via de-categorification

Abstract

Let $\mathfrak{g}$ be a Lie algebra over an algebraically closed field $\Bbbk$ 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 $\rho^*$ (i.e. the kernel of $\pi$ in the enveloping algebra $U(\mathfrak{g})$ contains that of $\rho^*$), and one relation $g_{\rho} = g_{\rho'}g_{\rho"}$ whenever $\rho$ is weakly contained in $\rho'\otimes\rho"$. The main result is that attaching to an irreducible representation its central character gives an isomorphism between $\mathcal{C}(\mathfrak{g})$ and the dual $\mathfrak{z}^*$ 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 $\mathfrak{sl}(\infty)$, $\mathfrak{o}(\infty)$ and $\mathfrak{sp}(\infty)$). These are analogues of a result of M\"uger'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.