Galois cohomology of algebraic groups acting on the tensor product of two composition algebras

TL;DR

This paper classifies certain algebraic structures formed from tensor products of composition algebras, analyzes their automorphisms, and extends cohomological invariant classifications to higher-dimensional Spin groups.

Contribution

It provides a comprehensive classification of algebras with involution arising from tensor products of composition algebras, including their invariants and automorphism groups, especially for bi-octonion algebras.

Findings

Classification of algebras with involution over tensor products of composition algebras.

Determination of automorphism, structure, and norm-similitude groups.

Extension of cohomological invariant classification to Spin(14).

Abstract

This is a study of algebras with involution that become isomorphic over a separable closure of the base field to a tensor product of two composition algebras. We classify these algebras, provide criteria for isomorphism and isotopy, and determine their automorphism groups, structure groups, and norm-similitude groups. The most interesting cases of these algebras are the $64$-dimensional bi-octonion algebras, on which groups of absolute type $(G_2 \times G_2)\rtimes \mathbb{Z}/2\mathbb{Z}$ and $\mathbf{Spin}_{14}$ act faithfully by automorphisms and norm-preserving isotopies, respectively. We classify the cohomological invariants of these algebras, characterise when they are division algebras, and study their 14-dimensional Albert forms and 64-dimensional octic norms. The main application that we reach is a classification of the mod 2 cohomological invariants of 14-dimensional quadratic forms in $I^3$, and of the group $\mathbf{Spin}_{14}$. This extends Garibaldi's classification of cohomological invariants for lower-dimensional Spin groups.