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Free Group Algebras in Division Rings with Valuation II

Part of: Rings and algebras with additional structure Conditions on elements Rings and algebras arising under various constructions Division rings and semisimple Artin rings

Published online by Cambridge University Press:  05 July 2019

Javier Sánchez
Javier Sánchez*
Affiliation:
Department of Mathematics - IME, University of São Paulo, Rua do Matão 1010, São Paulo, SP, 05508-090, Brazil Email: [email protected]
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Abstract

We apply the filtered and graded methods developed in earlier works to find (noncommutative) free group algebras in division rings.

If $L$ is a Lie algebra, we denote by $U(L)$ its universal enveloping algebra. P. M. Cohn constructed a division ring $\mathfrak{D}_{L}$ that contains $U(L)$. We denote by $\mathfrak{D}(L)$ the division subring of $\mathfrak{D}_{L}$ generated by $U(L)$.

Let $k$ be a field of characteristic zero, and let $L$ be a nonabelian Lie $k$-algebra. If either $L$ is residually nilpotent or $U(L)$ is an Ore domain, we show that $\mathfrak{D}(L)$ contains (noncommutative) free group algebras. In those same cases, if $L$ is equipped with an involution, we are able to prove that the free group algebra in $\mathfrak{D}(L)$ can be chosen generated by symmetric elements in most cases.

Let $G$ be a nonabelian residually torsion-free nilpotent group, and let $k(G)$ be the division subring of the Malcev–Neumann series ring generated by the group algebra $k[G]$. If $G$ is equipped with an involution, we show that $k(G)$ contains a (noncommutative) free group algebra generated by symmetric elements.

Keywords

division ringfiltered ringgraded ringvaluationring with involutionfree group algebrauniversal enveloping algebraordered group

MSC classification

Primary: 16K40: Infinite-dimensional and general
Secondary: 16W70: Filtered rings; filtrational and graded techniques 16W60: Valuations, completions, formal power series and related constructions 16W10: Rings with involution; Lie, Jordan and other nonassociative structures 16S10: Rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) 16S30: Universal enveloping algebras of Lie algebras 16U20: Ore rings, multiplicative sets, Ore localization
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 6 , December 2020 , pp. 1463 - 1504
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

Supported by FAPESP-Brazil, Proj. Temático 2015/09162-9, by Grant CNPq 307638/2015-4 and by MINECO-FEDER (Spain) through project numbers MTM2014-53644-P and MTM2017-83487-P.

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