Quaternion Algebras

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Quaternion Algebras

Author : John Voight
Publisher : Springer Nature
ISBN 13 : 3030566943
Total Pages : 877 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Quaternion Algebras by : John Voight

Download or read book Quaternion Algebras written by John Voight and published by Springer Nature. This book was released on 2021-06-28 with total page 877 pages. Available in PDF, EPUB and Kindle. Book excerpt: This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout.

Quaternions, Clifford Algebras and Relativistic Physics

Author : Patrick R. Girard
Publisher : Springer Science & Business Media
ISBN 13 : 3764377917
Total Pages : 180 pages
Book Rating : 4.7/5 (643 download)

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Book Synopsis Quaternions, Clifford Algebras and Relativistic Physics by : Patrick R. Girard

Download or read book Quaternions, Clifford Algebras and Relativistic Physics written by Patrick R. Girard and published by Springer Science & Business Media. This book was released on 2007-06-25 with total page 180 pages. Available in PDF, EPUB and Kindle. Book excerpt: The use of Clifford algebras in mathematical physics and engineering has grown rapidly in recent years. Whereas other developments have privileged a geometric approach, this book uses an algebraic approach that can be introduced as a tensor product of quaternion algebras and provides a unified calculus for much of physics. It proposes a pedagogical introduction to this new calculus, based on quaternions, with applications mainly in special relativity, classical electromagnetism, and general relativity.

Topics in Quaternion Linear Algebra

Author : Leiba Rodman
Publisher : Princeton University Press
ISBN 13 : 1400852749
Total Pages : 384 pages
Book Rating : 4.4/5 (8 download)

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Book Synopsis Topics in Quaternion Linear Algebra by : Leiba Rodman

Download or read book Topics in Quaternion Linear Algebra written by Leiba Rodman and published by Princeton University Press. This book was released on 2014-08-24 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations. Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.

Bilinear Algebra

Author : Kazimierz Szymiczek
Publisher : Routledge
ISBN 13 : 1351464205
Total Pages : 413 pages
Book Rating : 4.3/5 (514 download)

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Book Synopsis Bilinear Algebra by : Kazimierz Szymiczek

Download or read book Bilinear Algebra written by Kazimierz Szymiczek and published by Routledge. This book was released on 2017-11-22 with total page 413 pages. Available in PDF, EPUB and Kindle. Book excerpt: Giving an easily accessible elementary introduction to the algebraic theory of quadratic forms, this book covers both Witt's theory and Pfister's theory of quadratic forms. Leading topics include the geometry of bilinear spaces, classification of bilinear spaces up to isometry depending on the ground field, formally real fields, Pfister forms, the Witt ring of an arbitrary field (characteristic two included), prime ideals of the Witt ring, Brauer group of a field, Hasse and Witt invariants of quadratic forms, and equivalence of fields with respect to quadratic forms. Problem sections are included at the end of each chapter. There are two appendices: the first gives a treatment of Hasse and Witt invariants in the language of Steinberg symbols, and the second contains some more advanced problems in 10 groups, including the u-invariant, reduced and stable Witt rings, and Witt equivalence of fields.

Visualizing Quaternions

Author : Andrew J. Hanson
Publisher : Elsevier
ISBN 13 : 0080474772
Total Pages : 530 pages
Book Rating : 4.0/5 (84 download)

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Book Synopsis Visualizing Quaternions by : Andrew J. Hanson

Download or read book Visualizing Quaternions written by Andrew J. Hanson and published by Elsevier. This book was released on 2006-02-06 with total page 530 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduced 160 years ago as an attempt to generalize complex numbers to higher dimensions, quaternions are now recognized as one of the most important concepts in modern computer graphics. They offer a powerful way to represent rotations and compared to rotation matrices they use less memory, compose faster, and are naturally suited for efficient interpolation of rotations. Despite this, many practitioners have avoided quaternions because of the mathematics used to understand them, hoping that some day a more intuitive description will be available. The wait is over. Andrew Hanson's new book is a fresh perspective on quaternions. The first part of the book focuses on visualizing quaternions to provide the intuition necessary to use them, and includes many illustrative examples to motivate why they are important—a beautiful introduction to those wanting to explore quaternions unencumbered by their mathematical aspects. The second part covers the all-important advanced applications, including quaternion curves, surfaces, and volumes. Finally, for those wanting the full story of the mathematics behind quaternions, there is a gentle introduction to their four-dimensional nature and to Clifford Algebras, the all-encompassing framework for vectors and quaternions. Richly illustrated introduction for the developer, scientist, engineer, or student in computer graphics, visualization, or entertainment computing. Covers both non-mathematical and mathematical approaches to quaternions.

On Quaternions and Octonions

Author : John H. Conway
Publisher : CRC Press
ISBN 13 : 1439864187
Total Pages : 172 pages
Book Rating : 4.4/5 (398 download)

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Book Synopsis On Quaternions and Octonions by : John H. Conway

Download or read book On Quaternions and Octonions written by John H. Conway and published by CRC Press. This book was released on 2003-01-23 with total page 172 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book investigates the geometry of quaternion and octonion algebras. Following a comprehensive historical introduction, the book illuminates the special properties of 3- and 4-dimensional Euclidean spaces using quaternions, leading to enumerations of the corresponding finite groups of symmetries. The second half of the book discusses the less f

Quaternions and Cayley Numbers

Author : J.P. Ward
Publisher : Springer Science & Business Media
ISBN 13 : 9401157685
Total Pages : 252 pages
Book Rating : 4.4/5 (11 download)

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Book Synopsis Quaternions and Cayley Numbers by : J.P. Ward

Download or read book Quaternions and Cayley Numbers written by J.P. Ward and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: In essence, this text is written as a challenge to others, to discover significant uses for Cayley number algebra in physics. I freely admit that though the reading of some sections would benefit from previous experience of certain topics in physics - particularly relativity and electromagnetism - generally the mathematics is not sophisticated. In fact, the mathematically sophisticated reader, may well find that in many places, the rather deliberate progress too slow for their liking. This text had its origin in a 90-minute lecture on complex numbers given by the author to prospective university students in 1994. In my attempt to develop a novel approach to the subject matter I looked at complex numbers from an entirely geometric perspective and, no doubt in line with innumerable other mathematicians, re-traced steps first taken by Hamilton and others in the early years of the nineteenth century. I even enquired into the possibility of using an alternative multiplication rule for complex numbers (in which argzlz2 = argzl- argz2) other than the one which is normally accepted (argzlz2 = argzl + argz2). Of course, my alternative was rejected because it didn't lead to a 'product' which had properties that we now accept as fundamental (i. e.

The Book of Involutions

Author : Max-Albert Knus
Publisher : American Mathematical Soc.
ISBN 13 : 9780821873212
Total Pages : 624 pages
Book Rating : 4.8/5 (732 download)

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Book Synopsis The Book of Involutions by : Max-Albert Knus

Download or read book The Book of Involutions written by Max-Albert Knus and published by American Mathematical Soc.. This book was released on 1998-06-30 with total page 624 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is an exposition of the theory of central simple algebras with involution, in relation to linear algebraic groups. It provides the algebra-theoretic foundations for much of the recent work on linear algebraic groups over arbitrary fields. Involutions are viewed as twisted forms of (hermitian) quadrics, leading to new developments on the model of the algebraic theory of quadratic forms. In addition to classical groups, phenomena related to triality are also discussed, as well as groups of type $F_4$ or $G_2$ arising from exceptional Jordan or composition algebras. Several results and notions appear here for the first time, notably the discriminant algebra of an algebra with unitary involution and the algebra-theoretic counterpart to linear groups of type $D_4$. This volume also contains a Bibliography and Index. Features: original material not in print elsewhere a comprehensive discussion of algebra-theoretic and group-theoretic aspects extensive notes that give historical perspective and a survey on the literature rational methods that allow possible generalization to more general base rings

Quaternion Orders, Quadratic Forms, and Shimura Curves

Author : Montserrat Alsina
Publisher : American Mathematical Soc.
ISBN 13 : 9780821833599
Total Pages : 232 pages
Book Rating : 4.8/5 (335 download)

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Book Synopsis Quaternion Orders, Quadratic Forms, and Shimura Curves by : Montserrat Alsina

Download or read book Quaternion Orders, Quadratic Forms, and Shimura Curves written by Montserrat Alsina and published by American Mathematical Soc.. This book was released on 2004 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: Shimura curves are a far-reaching generalization of the classical modular curves. They lie at the crossroads of many areas, including complex analysis, hyperbolic geometry, algebraic geometry, algebra, and arithmetic. This monograph presents Shimura curves from a theoretical and algorithmic perspective. The main topics are Shimura curves defined over the rational number field, the construction of their fundamental domains, and the determination of their complex multiplicationpoints. The study of complex multiplication points in Shimura curves leads to the study of families of binary quadratic forms with algebraic coefficients and to their classification by arithmetic Fuchsian groups. In this regard, the authors develop a theory full of new possibilities that parallels Gauss'theory on the classification of binary quadratic forms with integral coefficients by the action of the modular group. This is one of the few available books explaining the theory of Shimura curves at the graduate student level. Each topic covered in the book begins with a theoretical discussion followed by carefully worked-out examples, preparing the way for further research.

Algebra

Author : Falko Lorenz
Publisher : Springer Science & Business Media
ISBN 13 : 0387724885
Total Pages : 343 pages
Book Rating : 4.3/5 (877 download)

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Book Synopsis Algebra by : Falko Lorenz

Download or read book Algebra written by Falko Lorenz and published by Springer Science & Business Media. This book was released on 2007-11-24 with total page 343 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is Volume II of a two-volume introductory text in classical algebra. The text moves methodically with numerous examples and details so that readers with some basic knowledge of algebra can read it without difficulty. It is recommended either as a textbook for some particular algebraic topic or as a reference book for consultations in a selected fundamental branch of algebra. The book contains a wealth of material. Amongst the topics covered in Volume are the theory of ordered fields and Nullstellen Theorems. Known researcher Lorenz also includes the fundamentals of the theory of quadratic forms, of valuations, local fields and modules. What’s more, the book contains some lesser known or nontraditional results – for instance, Tsen's results on the solubility of systems of polynomial equations with a sufficiently large number of indeterminates.

Finite-Dimensional Division Algebras over Fields

Author : Nathan Jacobson
Publisher : Springer Science & Business Media
ISBN 13 : 3642024297
Total Pages : 290 pages
Book Rating : 4.6/5 (42 download)

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Book Synopsis Finite-Dimensional Division Algebras over Fields by : Nathan Jacobson

Download or read book Finite-Dimensional Division Algebras over Fields written by Nathan Jacobson and published by Springer Science & Business Media. This book was released on 2009-12-09 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: Here, the eminent algebraist, Nathan Jacobsen, concentrates on those algebras that have an involution. Although they appear in many contexts, these algebras first arose in the study of the so-called "multiplication algebras of Riemann matrices". Of particular interest are the Jordan algebras determined by such algebras, and thus their structure is discussed in detail. Two important concepts also dealt with are the universal enveloping algebras and the reduced norm. However, the largest part of the book is the fifth chapter, which focuses on involutorial simple algebras of finite dimension over a field.

Dual Quaternions and Their Associated Clifford Algebras

Author : Ronald Goldman
Publisher : CRC Press
ISBN 13 : 1000903524
Total Pages : 215 pages
Book Rating : 4.0/5 (9 download)

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Book Synopsis Dual Quaternions and Their Associated Clifford Algebras by : Ronald Goldman

Download or read book Dual Quaternions and Their Associated Clifford Algebras written by Ronald Goldman and published by CRC Press. This book was released on 2023-09-29 with total page 215 pages. Available in PDF, EPUB and Kindle. Book excerpt: Clifford algebra for dual quaternions has emerged recently as an alternative to standard matrix algebra as a computational framework for computer graphics. This book presents dual quaternions and their associated Clifford algebras in a new light, accessible to and geared toward the computer graphics community. Collecting all the associated formulas and theorems in one place, this book provides an extensive and rigorous treatment of dual quaternions, as well as showing how two models of Clifford algebra emerge naturally from the theory of dual quaternions. Each section comes complete with a set of exercises to help readers sharpen and practice their understanding. This book is accessible to anyone with a basic knowledge of quaternion algebra and is of particular use to forward-thinking members of the computer graphics community.

Introduction to Quadratic Forms over Fields

Author : Tsit-Yuen Lam
Publisher : American Mathematical Soc.
ISBN 13 : 0821810952
Total Pages : 577 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Introduction to Quadratic Forms over Fields by : Tsit-Yuen Lam

Download or read book Introduction to Quadratic Forms over Fields written by Tsit-Yuen Lam and published by American Mathematical Soc.. This book was released on 2005 with total page 577 pages. Available in PDF, EPUB and Kindle. Book excerpt: This new version of the author's prizewinning book, Algebraic Theory of Quadratic Forms (W. A. Benjamin, Inc., 1973), gives a modern and self-contained introduction to the theory of quadratic forms over fields of characteristic different from two. Starting with few prerequisites beyond linear algebra, the author charts an expert course from Witt's classical theory of quadratic forms, quaternion and Clifford algebras, Artin-Schreier theory of formally real fields, and structural theorems on Witt rings, to the theory of Pfister forms, function fields, and field invariants. These main developments are seamlessly interwoven with excursions into Brauer-Wall groups, local and global fields, trace forms, Galois theory, and elementary algebraic K-theory, to create a uniquely original treatment of quadratic form theory over fields. Two new chapters totaling more than 100 pages have been added to the earlier incarnation of this book to take into account some of the newer results and more recent viewpoints in the area. As is characteristic of this author's expository style, the presentation of the main material in this book is interspersed with a copious number of carefully chosen examples to illustrate the general theory. This feature, together with a rich stock of some 280 exercises for the thirteen chapters, greatly enhances the pedagogical value of this book, both as a graduate text and as a reference work for researchers in algebra, number theory, algebraic geometry, algebraic topology, and geometric topology.

Rotations, Quaternions, and Double Groups

Author : Simon L. Altmann
Publisher : Courier Corporation
ISBN 13 : 0486317730
Total Pages : 315 pages
Book Rating : 4.4/5 (863 download)

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Book Synopsis Rotations, Quaternions, and Double Groups by : Simon L. Altmann

Download or read book Rotations, Quaternions, and Double Groups written by Simon L. Altmann and published by Courier Corporation. This book was released on 2013-04-09 with total page 315 pages. Available in PDF, EPUB and Kindle. Book excerpt: This self-contained text presents a consistent description of the geometric and quaternionic treatment of rotation operators, employing methods that lead to a rigorous formulation and offering complete solutions to many illustrative problems. Geared toward upper-level undergraduates and graduate students, the book begins with chapters covering the fundamentals of symmetries, matrices, and groups, and it presents a primer on rotations and rotation matrices. Subsequent chapters explore rotations and angular momentum, tensor bases, the bilinear transformation, projective representations, and the geometry, topology, and algebra of rotations. Some familiarity with the basics of group theory is assumed, but the text assists students in developing the requisite mathematical tools as necessary.

Quaternion Orders, Quadratic Forms, and Shimura Curves

Author : Montserrat Alsina and Pilar Bayer
Publisher : American Mathematical Soc.
ISBN 13 : 0821869833
Total Pages : 216 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Quaternion Orders, Quadratic Forms, and Shimura Curves by : Montserrat Alsina and Pilar Bayer

Download or read book Quaternion Orders, Quadratic Forms, and Shimura Curves written by Montserrat Alsina and Pilar Bayer and published by American Mathematical Soc.. This book was released on with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt: Shimura curves are a far-reaching generalization of the classical modular curves. They lie at the crossroads of many areas, including complex analysis, hyperbolic geometry, algebraic geometry, algebra, and arithmetic. This monograph presents Shimura curves from a theoretical and algorithmic perspective. The main topics are Shimura curves defined over the rational number field, the construction of their fundamental domains, and the determination of their complex multiplication points. The study of complex multiplication points in Shimura curves leads to the study of families of binary quadratic forms with algebraic coefficients and to their classification by arithmetic Fuchsian groups. In this regard, the authors develop a theory full of new possibilities that parallels Gauss' theory on the classification of binary quadratic forms with integral coefficients by the action of the modular group. This is one of the few available books explaining the theory of Shimura curves at the graduate student level. Each topic covered in the book begins with a theoretical discussion followed by carefully worked-out examples, preparing the way for further research. Titles in this series are co-published with the Centre de Recherches Mathématiques.

Central Simple Algebras and Galois Cohomology

Author : Philippe Gille
Publisher : Cambridge University Press
ISBN 13 : 1108293670
Total Pages : 432 pages
Book Rating : 4.1/5 (82 download)

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Book Synopsis Central Simple Algebras and Galois Cohomology by : Philippe Gille

Download or read book Central Simple Algebras and Galois Cohomology written by Philippe Gille and published by Cambridge University Press. This book was released on 2017-08-10 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields, this book starts from the basics and reaches such advanced results as the Merkurjev–Suslin theorem, a culmination of work initiated by Brauer, Noether, Hasse and Albert, and the starting point of current research in motivic cohomology theory by Voevodsky, Suslin, Rost and others. Assuming only a solid background in algebra, the text covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi–Brauer varieties, and techniques in Milnor K-theory and K-cohomology, leading to a full proof of the Merkurjev–Suslin theorem and its application to the characterization of reduced norms. The final chapter rounds off the theory by presenting the results in positive characteristic, including the theorems of Bloch–Gabber–Kato and Izhboldin. This second edition has been carefully revised and updated, and contains important additional topics.

Maximal Nilpotent Subalgebras I: Nilradicals and Cartan Subalgebras in Associative Algebras. With 428 Exercises

Author : Sven Bodo Wirsing
Publisher : Anchor Academic Publishing
ISBN 13 : 3960671032
Total Pages : 245 pages
Book Rating : 4.9/5 (66 download)

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Book Synopsis Maximal Nilpotent Subalgebras I: Nilradicals and Cartan Subalgebras in Associative Algebras. With 428 Exercises by : Sven Bodo Wirsing

Download or read book Maximal Nilpotent Subalgebras I: Nilradicals and Cartan Subalgebras in Associative Algebras. With 428 Exercises written by Sven Bodo Wirsing and published by Anchor Academic Publishing. This book was released on 2016-12 with total page 245 pages. Available in PDF, EPUB and Kindle. Book excerpt: During the author’s doctorate time at the Christian-Albrechts-Universitat to Kiel, Salvatore Siciliano gave a stimulating talk in the upper seminar algebra theory about Cartan subalgebras in Lie algebra associates to associative algebra. This talk was the incentive for the author to analyze maximal nilpotent substructures of the Lie algebra associated to associative algebras. In the present work Siciliano's theory about Cartan subalgebras is worked off and expanded to different special associative algebra classes. In addition, a second maximal nilpotent substructure is analyzed: the nilradical. Within this analysis the main focus is to describe these substructure with the associative structure of the underlying algebra. This is successfully realized in this work. Numerous examples (like group algebras and Solomon (Tits-) algebras) illustrate the results to the reader. Within the numerous exercises these results can be applied by the reader to get a deeper insight in this theory.