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• W2810229013 abstract "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." @default.
• W2810229013 created "2018-07-10" @default.
• W2810229013 creator A5011422867 @default.
• W2810229013 date "2019-07-05" @default.
• W2810229013 modified "2023-10-16" @default.
• W2810229013 title "Free Group Algebras in Division Rings with Valuation II" @default.
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• W2810229013 doi "https://doi.org/10.4153/s0008414x19000348" @default.
• W2810229013 hasPublicationYear "2019" @default.
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