Albert Algebras over Commutative Rings: The Last Frontier of Jordan Systems [Hardcover]

The first comprehensive book on Albert algebras, the Jordan algebras that are most important for Lie algebras and exceptional groups.This comprehensive reference on Albert algebras, the Jordan algebras most important for Lie algebras and exceptional groups, is the first of its kind. It is ideal for researchers in mathematics and physics interested in exceptional groups or non-associative algebras.This comprehensive reference on Albert algebras, the Jordan algebras most important for Lie algebras and exceptional groups, is the first of its kind. It is ideal for researchers in mathematics and physics interested in exceptional groups or non-associative algebras.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 inl$