Albert Algebras over Commutative Rings
The Last Frontier of Jordan Systems

The first comprehensive book on Albert algebras, the Jordan algebras that are most important for Lie algebras and exceptional groups.

Skip Garibaldi (Author), Holger P. Petersson (Author), Michel L. Racine (Author)

9781009426855, Cambridge University Press

Hardback, published 21 November 2024

678 pages
23.5 x 15.9 x 4.1 cm, 1.09 kg

Albert algebras provide key tools for understanding exceptional groups and related structures such as symmetric spaces. This self-contained book provides the first comprehensive reference on Albert algebras over fields without any restrictions on the characteristic of the field. As well as covering results in characteristic 2 and 3, many results are proven for Albert algebras over an arbitrary commutative ring, showing that they hold in this greater generality. The book extensively covers requisite knowledge, such as non-associative algebras over commutative rings, scalar extensions, projective modules, alternative algebras, and composition algebras over commutative rings, with a special focus on octonion algebras. It then goes into Jordan algebras, Lie algebras, and group schemes, providing exercises so readers can apply concepts. This centralized resource illuminates the interplay between results that use only the structure of Albert algebras and those that employ theorems about group schemes, and is ideal for mathematics and physics researchers.

I. Prologue: the ancient protagonists
II. Foundations
III. Alternative algebras
IV. Composition algebras
V. Jordan algebras
VI. Cubic Jordan algebras
VII. The two Tits constructions
VIII. Lie algebras
IX. Group schemes
References
Index of notation
Subject index.

Subject Areas: Algebra [PBF]