Free group algebras in division rings with valuation II

TL;DR

This paper demonstrates the existence of free group algebras within division rings associated with Lie algebras and groups, using filtered and graded methods, especially when certain algebraic conditions are met.

Contribution

It extends previous methods to show free group algebras exist in division rings linked to Lie algebras and groups under specific conditions, including involution symmetry.

Findings

Free group algebras exist in division rings for residually nilpotent Lie algebras.

Free group algebras can be generated by symmetric elements with involution.

Existence of free group algebras in division rings of certain residually torsion-free nilpotent groups.

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 $L$ 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 $k(G)$ 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.