A Taste of Jordan Algebras

 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.

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: 1977–1983.- 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.