On unitary groups associated to division algebras of degree three

TL;DR

This paper investigates the structure of special unitary groups linked to degree three division algebras with involutions, revealing the absence of hermitian elements and trivial intersections with certain subfields, advancing understanding of their algebraic properties.

Contribution

It demonstrates that these unitary groups lack hermitian and skew-hermitian elements, and their intersections with maximal subfields are trivial, providing new insights into their algebraic structure.

Findings

No hermitian or skew-hermitian elements in the groups.

Intersections with maximal subfields are trivial.

No reflections exist in these groups.

Abstract

We show that the special unitary group associated to an involution of the second kind on a central division algebra of degree three does not contain hermitian or skew-hermitian elements. Especially, there are no reflections. For Albert's special cyclic presentation we show that the intersections of reasonable $S$-arithmetic subgroups with the two obvious maximal subfields consist of the trivial central elements.