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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Bibliographic Information
- Alexandru Chirvasitu
- Affiliation: Department of Mathematics, University at Buffalo, Buffalo, New York 14260-2900
- MR Author ID: 868724
- Email: achirvas@buffalo.edu
- Received by editor(s): March 24, 2023
- Received by editor(s) in revised form: July 22, 2023, July 27, 2023, and September 3, 2023
- Published electronically: November 21, 2023
- Additional Notes: This work was partially supported by NSF grant DMS-2001128.
- © Copyright 2023 American Mathematical Society
- Journal: Represent. Theory 27 (2023), 1102-1125
- MSC (2020): Primary 22E27, 22E25, 22E15, 22E41, 22D10, 22E60, 20G20
- DOI: https://doi.org/10.1090/ert/662
- MathSciNet review: 4670402