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

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.

Skip Garibaldi is Director of the IDA Center for Communications Research, La Jolla. His research focuses on algebraic groups, especially exceptional groups, which naturally involves studying Albert algebras. He is co-author of the book 'Cohomological Invariants in Galois Cohomology (2003),' written with Alexander Merkurjev and Jean-Pierre Serre. He is a Fellow of the American Mathematical Society and received the Lester R. Ford Award from the Mathematical Association of America.

Holger P. Petersson is Emeritus Professor in the Faculty of Mathematics and Computer Science at FernUniversität in Hagen, Germany. He has published on non-associative algebras, especially Albert algebras, over the course of his whole career, and is the author of a key survey on Albert algebras over fields of arbitrary characteristic, which was a starting point for this book. He co-wrote the monograph 'Descent in Buildings' with Bernhard Mühlherr and Richard Weiss.

Michel L. Racine is Emeritus Professor in the Department of Mathematics and Statistics at the University of Ottawa. His research interests include Jordan algebras and superalgebras, especially Albert algebras. He has proved many key results in this book, either alone or in joint work with Petersson. He was a von Humboldt Fellow at Universität Münster and an SSHN Fellow at Université Paris 5.