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Cover: 9780387954479 | A Taste of Jordan Algebras | Kevin Mccrimmon | Buch | Universitext
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A Taste of Jordan Algebras
Buch von Kevin Mccrimmon
Sprache: Englisch

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Beschreibung
On several occasions I and colleagues have found ourselves teaching a o- semester course for students at the second year of graduate study in ma- ematics who want to gain a general perspective on Jordan algebras, their structure, and their role in mathematics, or want to gain direct experience with nonassociative algebra. These students typically have a solid grounding in ?rst¿year graduate algebra and the Artin¿Wedderburn theory of assoc- tive algebras, and a few have been introduced to Lie algebras (perhaps even Cayley algebras, in an o?hand way), but otherwise they have not seen any nonassociative algebras. Most of them will not go on to do research in non- sociative algebra, so the course is not primarily meant to be a training or breeding ground for research, though the instructor often hopes that one or two will be motivated to pursue the subject further. This text is meant to serve as an accompaniment to such a course. It is designed ?rst and foremost to be read by students on their own without assistance by a teacher. It is a direct mathematical conversation between the author and a reader whose mind (as far as nonassociative algebra goes) is a tabula rasa. In keeping with the tone of a private conversation, I give more heuristicandexplanatorycommentthanisusualingraduatetextsatthislevel (pep talks, philosophical pronouncements on the proper way to think about certain concepts, historical anecdotes, mention of some mathematicians who have contributed to our understanding of Jordan algebras, etc.
On several occasions I and colleagues have found ourselves teaching a o- semester course for students at the second year of graduate study in ma- ematics who want to gain a general perspective on Jordan algebras, their structure, and their role in mathematics, or want to gain direct experience with nonassociative algebra. These students typically have a solid grounding in ?rst¿year graduate algebra and the Artin¿Wedderburn theory of assoc- tive algebras, and a few have been introduced to Lie algebras (perhaps even Cayley algebras, in an o?hand way), but otherwise they have not seen any nonassociative algebras. Most of them will not go on to do research in non- sociative algebra, so the course is not primarily meant to be a training or breeding ground for research, though the instructor often hopes that one or two will be motivated to pursue the subject further. This text is meant to serve as an accompaniment to such a course. It is designed ?rst and foremost to be read by students on their own without assistance by a teacher. It is a direct mathematical conversation between the author and a reader whose mind (as far as nonassociative algebra goes) is a tabula rasa. In keeping with the tone of a private conversation, I give more heuristicandexplanatorycommentthanisusualingraduatetextsatthislevel (pep talks, philosophical pronouncements on the proper way to think about certain concepts, historical anecdotes, mention of some mathematicians who have contributed to our understanding of Jordan algebras, etc.
Über den Autor

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.

Zusammenfassung
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.
Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2003
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Buch
Reihe: Universitext
Inhalt: xxv
563 S.
ISBN-13: 9780387954479
ISBN-10: 0387954473
Sprache: Englisch
Ausstattung / Beilage: HC runder Rücken kaschiert
Einband: Gebunden
Autor: Mccrimmon, Kevin
Hersteller: Springer US
Springer New York
Universitext
Maße: 241 x 160 x 37 mm
Von/Mit: Kevin Mccrimmon
Erscheinungsdatum: 14.11.2003
Gewicht: 1,045 kg
Artikel-ID: 102554649
Über den Autor

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.

Zusammenfassung
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.
Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2003
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Buch
Reihe: Universitext
Inhalt: xxv
563 S.
ISBN-13: 9780387954479
ISBN-10: 0387954473
Sprache: Englisch
Ausstattung / Beilage: HC runder Rücken kaschiert
Einband: Gebunden
Autor: Mccrimmon, Kevin
Hersteller: Springer US
Springer New York
Universitext
Maße: 241 x 160 x 37 mm
Von/Mit: Kevin Mccrimmon
Erscheinungsdatum: 14.11.2003
Gewicht: 1,045 kg
Artikel-ID: 102554649
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