Let $\mathbb{ G}$ be a Lie group with solvable connected component and finitely-generated component group and $\alpha\in H^2(\mathbb{ G},\mathbb{ S}^1)$ a cohomology class. We prove that if $(\mathbb{ G},\alpha)$ is of type I then the same holds for the finite central extensions of $\mathbb{ G}$. In particular, finite central extensions of type-I connected solvable Lie groups are again of type I. This is by contrast with the general case, whereby the type-I property does not survive under finite central extensions. We also show that ad-algebraic hulls of connected solvable Lie groups operate on these even when the latter are not simply connected, and give a group-theoretic characterization of the intersection of all Euclidean subgroups of a connected, simply-connected solvable group $\mathbb{ G}$ containing a given central subgroup of $\mathbb{ G}$.
Categories: math.GR math.FA math.RT