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The Book of Involutions
Max-Albert Knus, Eidgenössische Technische Hochschule, Zürich, Switzerland, Alexander Merkurjev, University of California, Los Angeles, CA, Markus Rost, Universität Regensburg, Germany, and Jean-Pierre Tignol, Université Catholique de Louvain, Louvain-la-Neuve, Belgium
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Colloquium Publications
1998; 593 pp; hardcover
Volume: 44
ISBN-10: 0-8218-0904-0
ISBN-13: 978-0-8218-0904-4
List Price: US$84 Member Price: US$67.20
Order Code: COLL/44

This monograph is an exposition of the theory of central simple algebras with involution, in relation to linear algebraic groups. It provides the algebra-theoretic foundations for much of the recent work on linear algebraic groups over arbitrary fields. Involutions are viewed as twisted forms of (hermitian) quadrics, leading to new developments on the model of the algebraic theory of quadratic forms. In addition to classical groups, phenomena related to triality are also discussed, as well as groups of type $$F_4$$ or $$G_2$$ arising from exceptional Jordan or composition algebras. Several results and notions appear here for the first time, notably the discriminant algebra of an algebra with unitary involution and the algebra-theoretic counterpart to linear groups of type $$D_4$$. This volume also contains a Bibliography and Index.

Features:

• original material not in print elsewhere
• a comprehensive discussion of algebra-theoretic and group-theoretic aspects
• extensive notes that give historical perspective and a survey on the literature
• rational methods that allow possible generalization to more general base rings

Graduate students and research mathematicians interested in central simple algebras, linear algebraic groups, nonabelian Galois cohomology, and composition or Jordan algebras.

Reviews

"It is not only the book of involutions', but also the book of the classical groups' ... a very welcome addition to the literature. The topics treated here have been the objects of very intensive research. The specialists felt the need of a reference book, and the beginners of a good introduction. Both of these needs are fulfilled by the `book of involutions' ... a very useful reference source ... these results are not yet published elsewhere ... very well-written ... In addition to being an excellent exposition of many basic results concerning algebras with involution and the classical groups, the book also contains many new ideas and new results, often due to the authors themselves. The topic is a very beautiful and vital one, object of intensive current research. This research is now made easier thanks to the impressive work of the four authors."

-- Zentralblatt MATH

"This volume is a compendious study of algebras with involution, a subject with many facets which becomes particularly interesting for central simple algebras. The book is excellently written, and the chapters on algebraic groups and Galois cohomology alone would make the book an ideal read for aspiring postgraduate students of an algebraic persuasion. In addition, there is plenty of material to enlighten even those of us who already know something about the subject. All in all, this book recommends itself to anyone who wants a thorough reference source, complete with an ample selection of enlightening exercises and historical notes, which deals with exceptional Jordan algebras, Clifford algebras and modules, Tits' algebras and algebraic groups in a modern manner."

-- Bulletin of the London Mathematical Society

"The book under review is an important work which records many of the significant advances in the theory of algebras with involution that have taken place in recent years. Much of the material has not previously appeared in a book before; indeed, there is a substantial amount that has not appeared anywhere before. There is an interesting selection of exercises for each chapter which cover many ancillary results not included in the body of the text. Additionally, there are carefully prepared and highly informative notes at the end of each chapter which give historical commentary and sources for the topics covered. Overall, this book is an outstanding achievement. It will be an indispensable reference for the specialist, and a challenging but highly rewarding introduction for the novice."

-- Mathematical Reviews

• Involutions and Hermitian forms
• Invariants of involutions
• Similitudes
• Algebras of degree four
• Algebras of degree three
• Algebraic groups
• Galois cohomology
• Composition and triality
• Cubic Jordan algebras
• Trialitarian central simple algebras
• Bibliography
• Index
• Notation