Albert algebras over \(\mathbb{Z}\) and other rings. (English) Zbl 1527.17017
The paper studies Jordan algebras. The authors are interested in the Albert algebras. The classification of the simple Jordan algebras is well known, and the Albert algebra is the only one of these which is exceptional (that is it does not come from an associative algebra by means of passing to the symmetric, Jordan product). Moreover Albert algebras are closely related to simple affine group schemes of type \(F_4\), \(E_6\), \(E_7\). The authors of the paper under review classify Albert algebras over the ring of the integers \(\mathbb{Z}\). In fact they reduce the classification to that of groups of type \(F_4\) given by B. Conrad [Panor. Synth. 46, 193–253 (2015; Zbl 1356.14033)]. They also describe various aspects of the structure of Albert algebras over semilocal rings. The authors take great care in describing the novelty of their results, and the differences in their approach to the study of Albert algebras compared to the classical ones. The paper is well written and will be of significant interest to people working on nonassociative algebras.
Reviewer: Plamen Koshlukov (Campinas)
MSC:
| 17C40 | Exceptional Jordan structures |
| 17C30 | Associated groups, automorphisms of Jordan algebras |
| 20G41 | Exceptional groups |
Citations:
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