Free symmetric group algebras in division rings generated by poly-orderable groups

TL;DR

This paper demonstrates that for nonabelian poly-orderable groups, the Hughes-free division ring of fractions contains symmetric elements that generate a free group subalgebra, extending the canonical involution.

Contribution

It establishes the extension of the canonical involution to the Hughes-free division ring and identifies symmetric elements forming a free group subalgebra.

Findings

Canonical involution extends to the Hughes-free division ring

Existence of symmetric elements generating a free group subalgebra

Results apply to nonabelian poly-orderable groups

Abstract

We show that the canonical involution on a nonabelian poly-orderable group G extends to the Hughes-free division ring of fractions D of the group algebra k[G] of G over a field k and that, with respect to this involution, D contains a pair of symmetric elements freely generating a free group subalgebra of D over k.