Transactions of the Moscow Mathematical Society

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ISSN 1547-738X(e) ISSN 0077-1554(p)

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
Posted: January 12, 2012
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Abstract | References | Similar Articles | Additional Information

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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