Published online by Cambridge University Press: 05 July 2019
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.
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.