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A Taste of Jordan Algebrasby Kevin McCrimmon
Synopses & Reviews
In this book, Kevin McCrimmon describes the history of Jordan Algebras and he describes in full mathematical detail the recent structure theory for Jordan algebras of arbitrary dimension due to Efim Zel'manov. To keep the exposition elementary, the structure theory is developed for linear Jordan algebras, though the modern quadratic methods are used throughout. Both the quadratic methods and the Zelmanov results go beyond the previous textbooks on Jordan theory, written in the 1960's and 1980's before the theory reached its final form. This book is intended for graduate students and for individuals wishing to learn more about Jordan algebras. No previous knowledge is required beyond the standard first-year graduate algebra course. General students of algebra can profit from exposure to nonassociative algebras, and students or professional mathematicians working in areas such as Lie algebras, differential geometry, functional analysis, or exceptional groups and geometry can also profit from acquaintance with the material. Jordan algebras crop up in many surprising settings and can be applied to a variety of mathematical areas. Kevin McCrimmon introduced the concept of a quadratic Jordan algebra and developed a structure theory of Jordan algebras over an arbitrary ring of scalars. He is a Professor of Mathematics at the University of Virginia and the author of more than 100 research papers.
Describing the history of Jordan algebras, this book explores in full mathematical detail the structure theory for Jordan algebras of arbitrary dimension due to Efim Zelmanov. The book is written to serve either as a text for a second-year graduate course, or for independent reading.
Kevin McCrimmon introduced the concept of a quadratic Jordan algebra and developed a structure theory of Jordan algebras over an arbitrary ring of scalars. He is a Professor of Mathematics at the University of Virginia and the author of more than 100 research papers.
This book describes the history of Jordan algebras and describes in full mathematical detail the recent structure theory for Jordan algebras of arbitrary dimension due to Efim Zel'manov. Jordan algebras crop up in many surprising settings, and find application to a variety of mathematical areas. No knowledge is required beyond standard first-year graduate algebra courses.
Table of Contents
Introduction.- A Brief Survey of Jordan Theory: Origin of the Species; The Search for the Exceptional Algebras; Was sind und was sollen die Jordan Algebren? Connections with Lie Algebras and Groups; The Jordan River; Connections with the Real World; Connections with the Complex World.- The Historical Perspective: An Historical Survey of Jordan Structure Theory: Jordan Algebras in Physical Antiquity; Jordan Algebras in the Algebraic Renaissance; Jordan Algebras in the Enlightenment; The Classical Theory; The Final Classical Formulation; The Classical Methods; The Russian Revolution: 1977-1983; Zelmanov's Exceptional Methods.- The Classical Theory: The Category of Jordan Algebras; The Category of Alternative Algebras; Three Special Examples; Jordan Algebras of Cubic Forms; Two Basic Principles; Invertibility; Isotopes; Peirce Decomposition; Off-Diagonal Rules; Peirce Consequences; Spin Coordinatization; Hermitian Coordinatization; Multiple Peirce Decomposition; Multiple Peirce Consequences; Hermitian Symmetries; The Coordinate Algebra; Jacobson Coordinatization; Von Neumann Regularity; Inner Simplicity; Capacity; Herstein-Kleinfeld-Osborn.
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