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Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

 

Arens-Michael envelopes, homological epimorphisms, and relatively quasi-free algebras


Author: Alexei Yul'evich Pirkovskii
Translated by: Alex Martsinkovsky
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 69 (2008).
Journal: Trans. Moscow Math. Soc. 2008, 27-104
MSC (2000): Primary 46M18
Published electronically: November 19, 2008
MathSciNet review: 2549445
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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.


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Additional Information

Alexei Yul'evich Pirkovskii
Affiliation: Russian Peoples’ Friendship University, 117198 Moscow, Russia
Email: pirkosha@online.ru; pirkosha@sci.pfu.edu.ru

DOI: http://dx.doi.org/10.1090/S0077-1554-08-00169-6
PII: S 0077-1554(08)00169-6
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
Article copyright: © Copyright 2008 American Mathematical Society