# Gradings on tensor products of composition algebras and on the Smirnov algebra

## TL;DR

This paper classifies group gradings on tensor products of Cayley and Hurwitz algebras, and on Smirnov algebras, revealing their automorphism group structures and providing a comprehensive understanding of their symmetries.

## Contribution

It introduces a complete classification of gradings on tensor products of Cayley and Hurwitz algebras and on Smirnov algebras, and establishes isomorphisms of their automorphism group schemes.

## Key findings

- Classified gradings on tensor products of Cayley and Hurwitz algebras.
- Proved automorphism group schemes of tensor powers of Cayley algebra are isomorphic.
- Classified gradings on Smirnov algebras using automorphism group scheme isomorphisms.

## Abstract

We give classifications of group gradings, up to equivalence and up to isomorphism, on the tensor product of a Cayley algebra $\mathcal{C}$ and a Hurwitz algebra over a field of characteristic different from 2. We also prove that the automorphism group schemes of $\mathcal{C}^{\otimes n}$ and $\mathcal{C}^n$ are isomorphic. On the other hand, we prove that the automorphism group schemes of a Smirnov algebra (a $35$-dimensional simple exceptional structurable algebra constructed from a Cayley algebra $\mathcal{C}$) and $\mathcal{C}$ are isomorphic. This is used to obtain classifications, up to equivalence and up to isomorphism, of the group gradings on Smirnov algebras.