Finite central extensions of type I

Chirvasitu, Alexandru

doi:10.1090/ert/662

Finite central extensions of type I
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by Alexandru Chirvasitu;

Represent. Theory 27 (2023), 1102-1125

DOI: https://doi.org/10.1090/ert/662

Published electronically: November 21, 2023

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Abstract:

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 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 $\mathbb {G}$ containing a given central subgroup of $\mathbb {G}$.

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