Introduction to Octonion and Other Non-Associative Algebras in Physics
Introduction to Octonion and Other Non-Associative Algebras in Physics

Author: Susumu Okubo

Publisher: Cambridge University Press

Published: 1995-08-03

Total Pages: 152

ISBN-13: 0521472156

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In this book, the author aims to familiarize researchers and graduate students in both physics and mathematics with the application of non-associative algebras in physics.Topics covered by the author range from algebras of observables in quantum mechanics, angular momentum and octonions, division algebra, triple-linear products and YangSHBaxter equations. The author also covers non-associative gauge theoretic reformulation of Einstein's general relativity theory and so on. Much of the material found in this book is not available in other standard works.

Non-Associative Algebra and Its Applications
Non-Associative Algebra and Its Applications

Author: Lev Sabinin

Publisher: CRC Press

Published: 2006-01-13

Total Pages: 558

ISBN-13: 9780824726690

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With contributions derived from presentations at an international conference, Non-Associative Algebra and Its Applications explores a wide range of topics focusing on Lie algebras, nonassociative rings and algebras, quasigroups, loops, and related systems as well as applications of nonassociative algebra to geometry, physics, and natural sciences. This book covers material such as Jordan superalgebras, nonassociative deformations, nonassociative generalization of Hopf algebras, the structure of free algebras, derivations of Lie algebras, and the identities of Albert algebra. It also includes applications of smooth quasigroups and loops to differential geometry and relativity.

Progress in Physics, vol. 1/2016
Progress in Physics, vol. 1/2016

Author: Dmitri Rabounski

Publisher: Infinite Study

Published:

Total Pages: 94

ISBN-13:

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The Journal on Advanced Studies in Theoretical and Experimental Physics, including Related Themes from Mathematics

Non-Associative and Non-Commutative Algebra and Operator Theory
Non-Associative and Non-Commutative Algebra and Operator Theory

Author: Cheikh Thiécoumbe Gueye

Publisher: Springer

Published: 2016-11-21

Total Pages: 254

ISBN-13: 3319329022

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Presenting the collaborations of over thirty international experts in the latest developments in pure and applied mathematics, this volume serves as an anthology of research with a common basis in algebra, functional analysis and their applications. Special attention is devoted to non-commutative algebras, non-associative algebras, operator theory and ring and module theory. These themes are relevant in research and development in coding theory, cryptography and quantum mechanics. The topics in this volume were presented at the Workshop on Non-Associative & Non-Commutative Algebra and Operator Theory, held May 23—25, 2014 at Cheikh Anta Diop University in Dakar, Senegal in honor of Professor Amin Kaidi. The workshop was hosted by the university's Laboratory of Algebra, Cryptology, Algebraic Geometry and Applications, in cooperation with the University of Almería and the University of Málaga. Dr. Kaidi's work focuses on non-associative rings and algebras, operator theory and functional analysis, and he has served as a mentor to a generation of mathematicians in Senegal and around the world.

Non-Associative Normed Algebras
Non-Associative Normed Algebras

Author: Miguel Cabrera García

Publisher: Cambridge University Press

Published: 2018-04-12

Total Pages: 759

ISBN-13: 1107043115

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The first systematic account of the basic theory of normed algebras, without assuming associativity. Sure to become a central resource.

Non-Associative Normed Algebras: Volume 2, Representation Theory and the Zel'manov Approach
Non-Associative Normed Algebras: Volume 2, Representation Theory and the Zel'manov Approach

Author: Miguel Cabrera García

Publisher: Cambridge University Press

Published: 2018-04-12

Total Pages: 759

ISBN-13: 1108570763

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This first systematic account of the basic theory of normed algebras, without assuming associativity, includes many new and unpublished results and is sure to become a central resource for researchers and graduate students in the field. This second volume revisits JB*-triples, covers Zel'manov's celebrated work in Jordan theory, proves the unit-free variant of the Vidav–Palmer theorem, and develops the representation theory of alternative C*-algebras and non-commutative JB*-algebras. This completes the work begun in the first volume, which introduced these algebras and discussed the so-called non-associative Gelfand–Naimark and Vidav–Palmer theorems. This book interweaves pure algebra, geometry of normed spaces, and infinite-dimensional complex analysis. Novel proofs are presented in complete detail at a level accessible to graduate students. The book contains a wealth of historical comments, background material, examples, and an extensive bibliography.

The Geometry Of The Octonions
The Geometry Of The Octonions

Author: Tevian Dray

Publisher: World Scientific

Published: 2015-04-08

Total Pages: 229

ISBN-13: 9814401838

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There are precisely two further generalizations of the real and complex numbers, namely, the quaternions and the octonions. The quaternions naturally describe rotations in three dimensions. In fact, all (continuous) symmetry groups are based on one of these four number systems. This book provides an elementary introduction to the properties of the octonions, with emphasis on their geometric structure. Elementary applications covered include the rotation groups and their spacetime generalization, the Lorentz group, as well as the eigenvalue problem for Hermitian matrices. In addition, more sophisticated applications include the exceptional Lie groups, octonionic projective spaces, and applications to particle physics including the remarkable fact that classical supersymmetry only exists in particular spacetime dimensions.

Non-Associative Normed Algebras: Volume 1, The Vidav–Palmer and Gelfand–Naimark Theorems
Non-Associative Normed Algebras: Volume 1, The Vidav–Palmer and Gelfand–Naimark Theorems

Author: Miguel Cabrera García

Publisher: Cambridge University Press

Published: 2014-07-31

Total Pages: 735

ISBN-13: 1139992775

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This first systematic account of the basic theory of normed algebras, without assuming associativity, includes many new and unpublished results and is sure to become a central resource for researchers and graduate students in the field. This first volume focuses on the non-associative generalizations of (associative) C*-algebras provided by the so-called non-associative Gelfand–Naimark and Vidav–Palmer theorems, which give rise to alternative C*-algebras and non-commutative JB*-algebras, respectively. The relationship between non-commutative JB*-algebras and JB*-triples is also fully discussed. The second volume covers Zel'manov's celebrated work in Jordan theory to derive classification theorems for non-commutative JB*-algebras and JB*-triples, as well as other topics. The book interweaves pure algebra, geometry of normed spaces, and complex analysis, and includes a wealth of historical comments, background material, examples and exercises. The authors also provide an extensive bibliography.

Clifford Algebras and their Applications in Mathematical Physics
Clifford Algebras and their Applications in Mathematical Physics

Author: Rafał Abłamowicz

Publisher: Springer Science & Business Media

Published: 2000

Total Pages: 500

ISBN-13: 9780817641825

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The first part of a two-volume set concerning the field of Clifford (geometric) algebra, this work consists of thematically organized chapters that provide a broad overview of cutting-edge topics in mathematical physics and the physical applications of Clifford algebras. algebras and their applications in physics. Algebraic geometry, cohomology, non-communicative spaces, q-deformations and the related quantum groups, and projective geometry provide the basis for algebraic topics covered. Physical applications and extensions of physical theories such as the theory of quaternionic spin, a projective theory of hadron transformation laws, and electron scattering are also presented, showing the broad applicability of Clifford geometric algebras in solving physical problems. Treatment of the structure theory of quantum Clifford algebras, the connection to logic, group representations, and computational techniques including symbolic calculations and theorem proving rounds out the presentation.

Clifford Algebras and their Applications in Mathematical Physics
Clifford Algebras and their Applications in Mathematical Physics

Author: Rafal Ablamowicz

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 470

ISBN-13: 1461213681

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The plausible relativistic physical variables describing a spinning, charged and massive particle are, besides the charge itself, its Minkowski (four) po sition X, its relativistic linear (four) momentum P and also its so-called Lorentz (four) angular momentum E # 0, the latter forming four trans lation invariant part of its total angular (four) momentum M. Expressing these variables in terms of Poincare covariant real valued functions defined on an extended relativistic phase space [2, 7J means that the mutual Pois son bracket relations among the total angular momentum functions Mab and the linear momentum functions pa have to represent the commutation relations of the Poincare algebra. On any such an extended relativistic phase space, as shown by Zakrzewski [2, 7], the (natural?) Poisson bracket relations (1. 1) imply that for the splitting of the total angular momentum into its orbital and its spin part (1. 2) one necessarily obtains (1. 3) On the other hand it is always possible to shift (translate) the commuting (see (1. 1)) four position xa by a four vector ~Xa (1. 4) so that the total angular four momentum splits instead into a new orbital and a new (Pauli-Lubanski) spin part (1. 5) in such a way that (1. 6) However, as proved by Zakrzewski [2, 7J, the so-defined new shifted four a position functions X must fulfill the following Poisson bracket relations: (1.