Arens–Michael envelopes, homological epimorphisms, and relatively quasi-free algebras
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Alexei Yul’evich Pirkovskii
Translated by: Alex Martsinkovsky - Trans. Moscow Math. Soc. 2008, 27-104
- DOI: https://doi.org/10.1090/S0077-1554-08-00169-6
- Published electronically: November 19, 2008
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Abstract:
We describe and investigate Arens–Michael envelopes of associative algebras and their homological properties. We also introduce and study analytic analogs of some classical ring-theoretic constructs: Ore extensions, Laurent extensions, and tensor algebras. For some finitely generated algebras, we explicitly describe their Arens–Michael envelopes as certain algebras of noncommutative power series, and we also show that the embeddings of such algebras in their Arens–Michael envelopes are homological epimorphisms (i.e., localizations in the sense of J. Taylor). For that purpose we introduce and study the concepts of relative homological epimorphism and relatively quasi-free algebra. The above results hold for multiparameter quantum affine spaces and quantum tori, quantum Weyl algebras, algebras of quantum $(2\times 2)$-matrices, and universal enveloping algebras of some Lie algebras of small dimensions.References
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Bibliographic Information
- Alexei Yul’evich Pirkovskii
- Affiliation: Russian Peoples’ Friendship University, 117198 Moscow, Russia
- Email: pirkosha@online.ru; pirkosha@sci.pfu.edu.ru
- Published electronically: November 19, 2008
- Additional Notes: The author was supported by the RFFI grant No. 05-01-00982 and No. 05-01-00001 and the President of Russia’s grant MK-2049.2004.1
- © Copyright 2008 American Mathematical Society
- Journal: Trans. Moscow Math. Soc. 2008, 27-104
- MSC (2000): Primary 46M18
- DOI: https://doi.org/10.1090/S0077-1554-08-00169-6
- MathSciNet review: 2549445