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On holomorphic reflexivity conditions for complex Lie groups

Part of: Special aspects of infinite or finite groups Hopf algebras, quantum groups and related topics Lie groups

Published online by Cambridge University Press:  30 September 2021

O. Yu. Aristov
O. Yu. Aristov*
Affiliation:
Obninsk, Russia (aristovoyu@inbox.ru)
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Abstract

We consider Akbarov's holomorphic version of the non-commutative Pontryagin duality for a complex Lie group. We prove, under the assumption that $G$ is a Stein group with finitely many components, that (1) the topological Hopf algebra of holomorphic functions on $G$ is holomorphically reflexive if and only if $G$ is linear; (2) the dual cocommutative topological Hopf algebra of exponential analytic functional on $G$ is holomorphically reflexive. We give a counterexample, which shows that the first criterion cannot be extended to the case of infinitely many components. Nevertheless, we conjecture that, in general, the question can be solved in terms of the Banach-algebra linearity of $G$.

Keywords

Arens-Michael envelopecomplex Lie groupdistorted subgroupholomorphic function of exponential typeholomorphic reflexivitylinearizertopological Hopf algebra

MSC classification

Primary: 22E10: General properties and structure of complex Lie groups
Secondary: 16T05: Hopf algebras and their applications 20F65: Geometric group theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 4 , November 2021 , pp. 800 - 821
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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