Finite central extensions of type I

Abstract

Let G \mathbb {G} be a Lie group with solvable connected component and finitely-generated component group and α ∈ H 2 ( G , S 1 ) \alpha \in H^2(\mathbb {G},\mathbb {S}^1) a cohomology class. We prove that if ( G , α ) (\mathbb {G},\alpha ) is of type I then the same holds for the finite central extensions of G \mathbb {G} . In particular, finite central extensions of type-I connected solvable Lie groups are again of type I. This is in contrast to 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 G \mathbb {G} containing a given central subgroup of G \mathbb {G} .