A Taste of Jordan Algebras

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. The book is written to serve either as a text for a 2nd year graduate course, or for independent reading, for students who need or wish to know a bit about Jordan algebras.

Includes supplementary material: sn.pub/extras

Autorentext

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.

Klappentext

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.

Inhalt
A Colloquial Survey of Jordan Theory.- A Colloquial Survey of Jordan Theory.- A Historical Survey of Jordan Structure Theory.- Jordan Algebras in Physical Antiquity: The Search for an Exceptional Setting for Quantum Mechanics.- Jordan Algebras in the Algebraic Renaissance: Finite-Dimensional Jordan Algebras over Algebraically Closed Fields.- Jordan Algebras in the Enlightenment: Finite-Dimensional Jordan Algebras over General Fields.- The Classical Theory: Jordan Algebras with Minimum Condition.- The Final Classical Formulation: Algebras with Capacity.- The Classical Methods: Cherchez les Division Idempotents.- The Russian Revolution: 19771983.- Zel'manov'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.- Inverses.- Isotopes.- Peirce Decomposition.- Off-Diagonal Rules.- Peirce Consequences.- Spin Coordinatization.- Hermitian Coordinatization.- Multiple Peirce Decompositions.- Multiple Peirce Consequences.- Hermitian Symmetries.- The Coordinate Algebra.- Jacobson Coordinatization.- Von Neumann Regularity.- Inner Simplicity.- Capacity.- Herstein-Kleinfeld-Osborn Theorem.- Osborn's Capacity 2 Theorem.- Classical Classification.- Zel'manov's Exceptional Theorem.- The Radical.- Begetting and Bounding Idempotents.- Bounded Spectra Beget Capacity.- Absorbers of Inner Ideals.- Primitivity.- The Primitive Heart.- Filters and Ultrafilters.- Ultraproducts.- The Final Argument.