Involutions on tensor products of quaternion algebras

TL;DR

This paper investigates the decompositions of totally decomposable algebras with involution, especially those involving split quaternion factors, and constructs examples where such decompositions are not possible.

Contribution

It introduces new examples of tensor product algebras with involution that cannot be decomposed into split quaternion factors, extending previous results.

Findings

Constructed examples of algebras with involution that lack certain decompositions.

Extended earlier results on algebra decompositions with involution.

Used gauge theory to analyze algebra structures.

Abstract

We study possible decompositions of totally decomposable algebras with involution, that is, tensor products of quaternion algebras with involution. In particular, we are interested in decompositions in which one or several factors are the split quaternion algebra $M_2(F)$, endowed with an orthogonal involution. Using the theory of gauges, developed by Tignol-Wadsworth, we construct examples of algebras isomorphic to a tensor product of quaternion algebras with $k$ split factors, endowed with an involution which is totally decomposable, but does not admit any decomposition with $k$ factors $M_2(F)$ with involution. This extends an earlier result of Sivatski where the algebra considered is of degree $8$ and index $4$, and endowed with some orthogonal involution.