Unitary SK1 of graded and valued division algebras, I

TL;DR

This paper studies the unitary SK1 group of graded division algebras with involution, showing that the graded approach simplifies calculations and extends results to broader classes over henselian fields.

Contribution

It demonstrates that calculations in the graded setting are simpler and establishes a link between graded and non-graded cases, enabling wider applicability.

Findings

Graded setting simplifies SK1 calculations.

Unitary SK1 of tame valued division algebra equals that of its graded algebra.

Extended SK1 computations to broader classes over henselian fields.

Abstract

The reduced unitary Whitehead group SK1 of a graded division algebra equipped with a unitary involution (i.e., an involution of the second kind) and graded by a torsion-free abelian group is studied. It is shown that calculations in the graded setting are much simpler than their nongraded counterparts. The bridge to the non-graded case is established by proving that the unitary SK1 of a tame valued division algebra wih a unitary involution over a henselian field coincides with the unitary SK1 of its associated graded division algebra. As a consequence, the graded approach allows us not only to recover results available in the literature with substantially easier proofs, but also to calculate the unitary SK1 for much wider classes of division algebras over henselian fields.