Comparing invariants of SK1

TL;DR

This paper compares various invariants of the SK1 group of central simple algebras, demonstrating their equivalences and non-triviality through explicit computations and generalizations.

Contribution

It establishes the equivalence of different invariants for biquaternion algebras and proves the non-triviality of Kahn's invariant, extending known formulas.

Findings

Invariants of SK1 for biquaternion algebras are essentially the same.

Kahn's invariant is proven to be non-trivial.

A generalized formula for the invariant on tensor products of symbol algebras.

Abstract

In this text, we compare several invariants of the reduced Whitehead group SK1 of a central simple algebra. For biquaternion algebras, we compare a generalised invariant of Suslin as constructed by the author in a previous article to an invariant introduced by Knus-Merkurjev-Rost-Tignol. Using explicit computations, we prove these invariants are essentially the same. We also prove the non-triviality of an invariant introduced by Kahn. To obtain this result, we compare Kahn's invariant to an invariant introduced by Suslin in 1991 which is non-trivial for Platonov's examples of non-trivial SK1. We also give a formula for the value on the centre of the tensor product of two symbol algebras which generalises a formula of Merkurjev for biquaternion algebras.