Geodesic
Free Group Algebras in Division Rings with Valuation II
Canadian journal of mathematics, Tome 72 (2020) no. 6, pp. 1463-1504

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/S0008414X19000348
Mots-clés : division ring, filtered ring, graded ring, valuation, ring with involution, free group algebra, universal enveloping algebra, ordered group 
Sánchez, Javier. Free Group Algebras in Division Rings with Valuation II. Canadian journal of mathematics, Tome 72 (2020) no. 6, pp. 1463-1504. doi: 10.4153/S0008414X19000348
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