Abstract
We prove that every locally bounded automorphism of a linear connected Lie central extension of a connected perfect Lie group is continuous if and only if it is continuous on the center. We also prove that, if \(Z\) is a connected Abelian group without nontrivial compact subgroups, \(H\) is a connected perfect Lie group and the short sequence of Lie groups \(\{e\}\to Z\to G\to H\to\{e\}\) is exact, then every locally bounded automorphism of \(G\) is continuous if and only if it is continuous on the center of \(G\).
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A. I. Shtern, “Locally Bounded Automorphisms of Connected Reductive Lie Groups”, Russ. J. Math. Phys., 28:3 (2021), 356–357.
V. S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Prentice-Hall Inc., Englewood Cliffs, NJ, 1974.
A. I. Shtern, “A Version of van der Waerden’s Theorem and a Proof of Mishchenko’s Conjecture on Homomorphisms of Locally Compact Groups”, Izv. Math., 72:1 (2008), 169–205.
A. I. Shtern, “Finite-Dimensional Quasirepresentations of Connected Lie Groups and Mishchenko’s Conjecture”, J. Math. Sci. (N. Y.), 159:5 (2009), 653–751.
A. I. Shtern, “Locally Bounded Finally Precontinuous Finite-Dimensional Quasirepresentations of Locally Compact Groups”, Sb. Math., 208:10 (2017), 1557–1576.
A. I. Shtern, “A Criterion for the Continuity with Respect to the Original Group Topology of the Restriction to the Commutator Subgroup for a Locally Bounded Finite-Dimensional Representation of a Connected Lie Group”, Proc. Jangjeon Math. Soc., 22:1 (2019), 201–204.
I. Namioka, “Separate Continuity and Joint Continuity”, Pacific J. Math., 51 (1974), 515–531.
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Partially supported by the Moscow Center for Fundamental and Applied Mathematics.
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Shtern, A.I. Continuity Criteria for Locally Bounded Automorphisms of Central Extensions of Perfect Lie Groups. Russ. J. Math. Phys. 28, 543–544 (2021). https://doi.org/10.1134/S1061920821040117
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DOI: https://doi.org/10.1134/S1061920821040117