Free symmetric and unitary pairs in division rings infinite-dimensional over their centers

TL;DR

This paper demonstrates the existence of free symmetric and unitary pairs in the multiplicative groups of certain infinite-dimensional division rings over their centers, with respect to a $k$-involution.

Contribution

It establishes the presence of free symmetric and unitary pairs in specific classes of infinite-dimensional division rings, expanding understanding of their algebraic structure.

Findings

Existence of free symmetric pairs in certain division rings

Existence of free unitary pairs in certain division rings

Results depend on the division ring belonging to specific families

Abstract

Let $D$ be a division ring infinite-dimensional over its center $k$ with multiplicative group $D^{\times}$. We show that if $D$ belongs to certain families, there exist free symmetric and unitary pairs in $D^{\times}$ with respect to a $k$-involution on $D$.