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

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.

💡 Research Summary

This document serves as a comprehensive companion to the 2024 Cambridge monograph Albert algebras over commutative rings by Garibaldi, Petersson, and Racine. It is organized into three main parts: errata, addenda, and detailed solutions to all exercises across the nine chapters of the book.

Errata – A page‑by‑page list of typographical and substantive mistakes is provided, each entry indicating the original location, the corrected text, and the date on which the correction was incorporated into the online draft. Corrections range from minor notation fixes (e.g., replacing “R” with “k” in the definition of the U‑operator) to more substantial clarifications (e.g., the characteristic‑3 hypothesis in the description of the first Tits construction). The errata ensure that readers can reliably cross‑reference the printed edition with the up‑to‑date electronic version.

Addenda – Four substantial extensions of the original material are presented, each labeled A1 through A4.

Addendum A1 proves that every Albert algebra over a non‑zero commutative ring is i‑exceptional, a stronger notion than the classical “exceptional” property. The authors introduce the Glennie polynomial (g_9(x,y,z)), a degree‑9, skew‑symmetric map originally studied in the context of special Jordan algebras. Lemma A1.6 shows that (g_9) does not vanish on a Hermitian (3\times3) matrix algebra (\operatorname{Her}_3(C)) when the underlying composition algebra (C) is non‑associative. Lemma A1.7 proves that (g_9) vanishes on every i‑special Jordan algebra (i.e., a homomorphic image of a special Jordan algebra). Combining these with a flat‑descent argument (Lemma A1.4) yields the main theorem: Albert algebras are i‑exceptional over any non‑zero base ring. As a corollary (Theorem A1.2) the classical classification of central simple Jordan algebras over fields is recovered: a central simple Jordan algebra is exceptional if and only if it is an Albert algebra.

Addendum A2 addresses the problem of descent: given a Jordan (or cubic Jordan) algebra defined over a field extension (F) of an integral domain (k), under what conditions does a (k)-submodule inherit the algebraic structure? Lemma A2.1 establishes a permanence principle for polynomial laws: if a polynomial law becomes zero after tensoring with some field containing (k), then it is already zero over (k). Using this, Proposition A2.2 shows that a para‑quadratic (k)-algebra whose base change to a field is a Jordan algebra is itself a Jordan algebra. Proposition A2.3 translates the result into lattice language: a finitely generated (k)-lattice (\Lambda) inside a Jordan algebra (J) over a field (F) becomes a Jordan algebra over (k) provided either (1) (\Lambda) is closed under the (U)-operator, or (2) (2) is invertible in (k) and (\Lambda) is closed under squaring. The authors then extend the discussion to cubic Jordan algebras of degree 3 over Dedekind domains, emphasizing projectivity of lattices and the compatibility of the cubic norm structure with descent.

Addendum A3 concerns splitting fields for Freudenthal algebras. The authors prove that any Freudenthal algebra of rank 9 (i.e., a cubic Jordan algebra associated with a composition algebra) becomes split over a field extension whose degree divides 6. The proof exploits the structure of Zorn algebras (\operatorname{Zor}(k)) and the associated cubic norm (N), showing that the invariants (n_C) and (t_C) become trivial after adjoining square‑roots and cube‑roots of suitable elements, leading to an extension of degree at most 6. This result generalizes earlier field‑splitting theorems that required characteristic‑different‑2 assumptions, providing a uniform bound valid in all characteristics.

Addendum A4 gives a Tits‑construction description of Albert algebras. Starting from a rank‑9 Freudenthal subalgebra (J_0) inside an Albert algebra (J), the authors construct a pair ((D,\mu)) where (D) is a central simple algebra of degree 3 with involution and (\mu) a scalar, such that (J) is isomorphic to the first Tits construction (J(D,\mu)). They also discuss the second Tits construction, showing how the same subalgebra can be used to recover the Albert algebra via a twisted composition algebra. Detailed formulas for the Jordan product, the cubic norm, and the trace are provided, together with explicit examples over (\mathbb{R}) and (\mathbb{C}).

Solutions – The bulk of the document consists of step‑by‑step solutions to every exercise in the book, organized by chapter and section. The solutions faithfully follow the notation and style of the original text, often expanding on hints given in the exercises. Highlights include:

• Chapter V: a proof that (\operatorname{Der}(J)) is a finitely generated projective module of rank 52 for any Albert algebra (J) over a principal ideal domain, using the identification (\operatorname{Lie}(\operatorname{Aut}(J))\cong \mathfrak{f}_4).
• Chapter VIII: verification that a certain subspace of the octonion algebra (R^8) satisfies the required Lie bracket identities, correcting a sign error in the printed version.
• Chapter IX: construction of group schemes of type (E_7) associated with Albert algebras, and a detailed analysis of the adjoint representation.

Overall, the document not only corrects the printed book but also enriches it with new theoretical insights, especially the notion of i‑exceptionality and the systematic use of the Glennie polynomial. It provides researchers and graduate students with a reliable reference for both the foundational material on Albert algebras and the latest developments in their structure theory, descent properties, splitting behavior, and connections to exceptional groups via Tits constructions.