  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.
