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Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

 

Connected locally compact groups: The Hochschild kernel and faithfulness of locally bounded finite-dimensional representations


Author: A. I. Shtern
Translated by: G. G. Gould
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 72 (2011), vypusk 1.
Journal: Trans. Moscow Math. Soc. 2011, 79-95
MSC (2010): Primary 22E15, 22C05
Published electronically: January 12, 2012
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Abstract: We obtain a number of consequences of the theorem on the automatic continuity of locally bounded finite-dimensional representations of connected Lie groups on the derived subgroup of the group, as well as an analogue of Lie's theorem for (not necessarily continuous) finite-dimensional representations of connected soluble locally compact groups. In particular, we give a description of connected Lie groups admitting a (not necessarily continuous) faithful locally bounded finite-dimensional representation; as it turns out, such groups are linear. Furthermore, we give a description of the intersection of the kernels of continuous finite-dimensional representations of a given connected locally compact group, obtain a generalization of Hochschild's theorem on the kernel of the universal representation in terms of locally bounded (not necessarily continuous) finite-dimensional linear representations, and find the intersection of the kernels of such representations for a connected reductive Lie group.


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Additional Information

A. I. Shtern
Affiliation: Faculty of Mechanics and Mathematics, Moscow State University, Russia
Email: ashtern@member.ams.org

DOI: http://dx.doi.org/10.1090/S0077-1554-2012-00190-3
PII: S 0077-1554(2012)00190-3
Keywords: Locally compact group, almost connected locally compact group, Freudenthal–Weil theorem, MAP group, semisimple locally compact group, locally bounded map.
Published electronically: January 12, 2012
Additional Notes: This work was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 08-01-000034) and the Programme of Support for Leading Scientific Schools (grant no. NSh-1562.2008.1)
Article copyright: © Copyright 2012 American Mathematical Society