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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Arens-Michael enveloping algebras and analytic smash products


Author: A. Yu. Pirkovskii
Journal: Proc. Amer. Math. Soc. 134 (2006), 2621-2631
MSC (2000): Primary 46M18, 46H05, 16S30, 16S40, 18G25
Published electronically: February 17, 2006
MathSciNet review: 2213741
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Abstract: Let $ \mathfrak{g}$ be a finite-dimensional complex Lie algebra, and let $ U(\mathfrak{g})$ be its universal enveloping algebra. We prove that if $ \widehat{U}(\mathfrak{g})$, the Arens-Michael envelope of $ U(\mathfrak{g})$ is stably flat over $ U(\mathfrak{g})$ (i.e., if the canonical homomorphism $ U(\mathfrak{g})\to\widehat{U}(\mathfrak{g})$ is a localization in the sense of Taylor (1972), then $ \mathfrak{g}$ is solvable. To this end, given a cocommutative Hopf algebra $ H$ and an $ H$-module algebra $ A$, we explicitly describe the Arens-Michael envelope of the smash product $ A\char93 H$ as an ``analytic smash product'' of their completions w.r.t. certain families of seminorms.


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Additional Information

A. Yu. Pirkovskii
Affiliation: Department of Nonlinear Analysis and Optimization, Faculty of Science, Peoples’ Friendship University of Russia, Mikluho-Maklaya 6, 117198 Moscow, Russia
Email: pirkosha@sci.pfu.edu.ru, pirkosha@online.ru

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08251-7
PII: S 0002-9939(06)08251-7
Received by editor(s): July 20, 2004
Received by editor(s) in revised form: March 24, 2005
Published electronically: February 17, 2006
Additional Notes: This work was partially supported by the RFBR grants 05-01-00982 and 05-01-00001, and by the President of Russia grant MK-2049.2004.1.
Communicated by: Martin Lorenz
Article copyright: © Copyright 2006 American Mathematical Society