Structure and Representations of Jordan Algebras
Structure and Representations of Jordan Algebras

Author: Nathan Jacobson

Publisher: American Mathematical Soc.

Published: 1968-12-31

Total Pages: 464

ISBN-13: 082184640X

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The theory of Jordan algebras has played important roles behind the scenes of several areas of mathematics. Jacobson's book has long been the definitive treatment of the subject. It covers foundational material, structure theory, and representation theory for Jordan algebras. Of course, there are immediate connections with Lie algebras, which Jacobson details in Chapter 8. Of particular continuing interest is the discussion of exceptional Jordan algebras, which serve to explain the exceptional Lie algebras and Lie groups. Jordan algebras originally arose in the attempts by Jordan, von Neumann, and Wigner to formulate the foundations of quantum mechanics. They are still useful and important in modern mathematical physics, as well as in Lie theory, geometry, and certain areas of analysis.

A Taste of Jordan Algebras
A Taste of Jordan Algebras

Author: Kevin McCrimmon

Publisher: Springer Science & Business Media

Published: 2006-05-29

Total Pages: 584

ISBN-13: 0387217967

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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.

Jordan Structures in Lie Algebras
Jordan Structures in Lie Algebras

Author: Antonio Fernández López

Publisher: American Mathematical Soc.

Published: 2019-08-19

Total Pages: 314

ISBN-13: 1470450860

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Explores applications of Jordan theory to the theory of Lie algebras. After presenting the general theory of nonassociative algebras and of Lie algebras, the book then explains how properties of the Jordan algebra attached to a Jordan element of a Lie algebra can be used to reveal properties of the Lie algebra itself.

The Geometry of Jordan and Lie Structures
The Geometry of Jordan and Lie Structures

Author: Wolfgang Bertram

Publisher: Springer

Published: 2003-07-01

Total Pages: 285

ISBN-13: 3540444580

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The geometry of Jordan and Lie structures tries to answer the following question: what is the integrated, or geometric, version of real Jordan algebras, - triple systems and - pairs? Lie theory shows the way one has to go: Lie groups and symmetric spaces are the geometric version of Lie algebras and Lie triple systems. It turns out that both geometries are closely related via a functor between them, called the Jordan-Lie functor, which is constructed in this book. The reader is not assumed to have any knowledge of Jordan theory; the text can serve as a self-contained introduction to (real finite-dimensional) Jordan theory.

Jordan Algebras in Analysis, Operator Theory, and Quantum Mechanics
Jordan Algebras in Analysis, Operator Theory, and Quantum Mechanics

Author: Harald Upmeier

Publisher: American Mathematical Soc.

Published: 1987

Total Pages: 95

ISBN-13: 082180717X

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Jordan algebras have found interesting applications in seemingly unrelated areas of mathematics such as operator theory, the foundations of quantum mechanics, complex analysis in finite and infinite dimensions, and harmonic analysis on homogeneous spaces. This book describes some relevant results and puts them in a general framework.

Jordan Algebras and Algebraic Groups
Jordan Algebras and Algebraic Groups

Author: Tonny A. Springer

Publisher: Springer Science & Business Media

Published: 1997-12-11

Total Pages: 202

ISBN-13: 9783540636328

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From the reviews: "This book presents an important and novel approach to Jordan algebras. [...] Springer's work will be of service to research workers familiar with linear algebraic groups who find they need to know something about Jordan algebras and will provide Jordan algebraists with new techniques and a new approach to finite-dimensional algebras over fields." American Scientist

Jordan Triple Systems by the Grid Approach
Jordan Triple Systems by the Grid Approach

Author: Erhard Neher

Publisher: Springer

Published: 2006-11-15

Total Pages: 206

ISBN-13: 354047921X

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Grids are special families of tripotents in Jordan triple systems. This research monograph presents a theory of grids including their classification and coordinization of their cover. Among the applications given are - classification of simple Jordan triple systems covered by a grid, reproving and extending most of the known classification theorems for Jordan algebras and Jordan pairs - a Jordan-theoretic interpretation of the geometry of the 27 lines on a cubic surface - structure theories for Hilbert-triples and JBW*-triples, the Jordan analogues of Hilbert-triples and W*-algebras which describe certain symmetric Banach manifolds. The notes are essentially self-contained and independent of the structure theory of Jordan algebras and Jordan pairs. They can be read by anyone with a basic knowledge in algebraic geometry or functional analysis. The book is intended to serve both as a reference for researchers in Jordan theory and as an introductory textbook for newcomers to the subject.

Octonions, Jordan Algebras and Exceptional Groups
Octonions, Jordan Algebras and Exceptional Groups

Author: Tonny A. Springer

Publisher: Springer

Published: 2013-12-21

Total Pages: 212

ISBN-13: 3662126222

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The 1963 Göttingen notes of T. A. Springer are well known in the field but have been unavailable for some time. This book is a translation of those notes, completely updated and revised. The part of the book dealing with the algebraic structures is on a fairly elementary level, presupposing basic results from algebra.