Solutions to the exercises from the book "Albert algebras over commutative rings"

TL;DR

This paper provides detailed solutions, errata, and addenda for exercises in a book on Albert algebras over commutative rings, including proofs of key properties and classifications of these algebras.

Contribution

It offers new proofs and clarifications of properties of Albert algebras, including their exceptionality, lattice structures, and relations to Freudenthal algebras, expanding understanding of their structure.

Findings

Albert algebras are exceptional and central simple Jordan algebras over fields are Albert algebras if and only if they are exceptional.

A regular lattice in a real Albert algebra is itself an Albert algebra.

A Freudenthal algebra over a field is split by an extension of degree dividing 6.

Abstract

This document presents the solutions to the exercises in the book "Albert algebras over commutative rings" published by Cambridge University Press, 2024, as well as errata and addenda. The addenda include proofs, in the style of the book, showing that (A1) Albert algebras are exceptional and in particular that a central simple Jordan algebra over a field is exceptional if and only if it is an Albert algebra; (A2) A regular lattice in a real Albert algebra is also an Albert algebra; (A3) a Freudenthal algebra over a field is split by an extension of degree dividing 6; and (A4) a Freudenthal subalgebra of rank 9 in an Albert algebra can be used to describe the Albert algebra as a Tits construction.