Arens-Michael enveloping algebras and analytic smash products
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- by A. Yu. Pirkovskii PDF
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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 $A\# H$ as an “analytic smash product” of their completions w.r.t. certain families of seminorms.References
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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
- 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
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 134 (2006), 2621-2631
- MSC (2000): Primary 46M18, 46H05, 16S30, 16S40, 18G25
- DOI: https://doi.org/10.1090/S0002-9939-06-08251-7
- MathSciNet review: 2213741