Lie-algebra centers via de-categorification

TL;DR

This paper introduces a new group associated with a Lie algebra that encodes its center via irreducible representations, establishing isomorphisms in certain cases and revealing when this group is trivial, thus generalizing known results.

Contribution

The paper defines the universal grading group for Lie algebras and proves it is isomorphic to the dual of the center for solvable and semisimple cases, extending center-reconstruction results.

Findings

Isomorphism between the universal grading group and the dual of the center for solvable Lie algebras.

Isomorphism between the universal grading group and the dual of the center for semisimple Lie algebras.

Universal grading group is trivial for Lie algebras with faithful irreducible representations, including certain infinite-dimensional algebras.

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.