Clifford Algebras and the Classical Groups

Clifford Algebras and the Classical Groups

Author: Ian R. Porteous

Publisher: Cambridge University Press

Published: 1995-10-05

Total Pages: 309

ISBN-13: 0521551773

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The Clifford algebras of real quadratic forms and their complexifications are studied here in detail, and those parts which are immediately relevant to theoretical physics are seen in the proper broad context. Central to the work is the classification of the conjugation and reversion anti-involutions that arise naturally in the theory. It is of interest that all the classical groups play essential roles in this classification. Other features include detailed sections on conformal groups, the eight-dimensional non-associative Cayley algebra, its automorphism group, the exceptional Lie group G(subscript 2), and the triality automorphism of Spin 8. The book is designed to be suitable for the last year of an undergraduate course or the first year of a postgraduate course.


Book Synopsis Clifford Algebras and the Classical Groups by : Ian R. Porteous

Download or read book Clifford Algebras and the Classical Groups written by Ian R. Porteous and published by Cambridge University Press. This book was released on 1995-10-05 with total page 309 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Clifford algebras of real quadratic forms and their complexifications are studied here in detail, and those parts which are immediately relevant to theoretical physics are seen in the proper broad context. Central to the work is the classification of the conjugation and reversion anti-involutions that arise naturally in the theory. It is of interest that all the classical groups play essential roles in this classification. Other features include detailed sections on conformal groups, the eight-dimensional non-associative Cayley algebra, its automorphism group, the exceptional Lie group G(subscript 2), and the triality automorphism of Spin 8. The book is designed to be suitable for the last year of an undergraduate course or the first year of a postgraduate course.

Classical Groups and Geometric Algebra

Classical Groups and Geometric Algebra

Author: Larry C. Grove

Publisher: American Mathematical Soc.

Published: 2002

Total Pages: 181

ISBN-13: 0821820192

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A graduate-level text on the classical groups: groups of matrices, or (more often) quotients of matrix groups by small normal subgroups. It pulls together into a single source the basic facts about classical groups defined over fields, together with the required geometrical background information, from first principles. The chief prerequisites are basic linear algebra and abstract algebra, including fundamentals of group theory and some Galois Theory. The author teaches at the U. of Arizona. c. Book News Inc.


Book Synopsis Classical Groups and Geometric Algebra by : Larry C. Grove

Download or read book Classical Groups and Geometric Algebra written by Larry C. Grove and published by American Mathematical Soc.. This book was released on 2002 with total page 181 pages. Available in PDF, EPUB and Kindle. Book excerpt: A graduate-level text on the classical groups: groups of matrices, or (more often) quotients of matrix groups by small normal subgroups. It pulls together into a single source the basic facts about classical groups defined over fields, together with the required geometrical background information, from first principles. The chief prerequisites are basic linear algebra and abstract algebra, including fundamentals of group theory and some Galois Theory. The author teaches at the U. of Arizona. c. Book News Inc.

Groups and Characters

Groups and Characters

Author: Larry C. Grove

Publisher: John Wiley & Sons

Published: 2011-09-26

Total Pages: 228

ISBN-13: 1118030931

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An authoritative, full-year course on both group theory and ordinary character theory--essential tools for mathematics and the physical sciences One of the few treatments available combining both group theory and character theory, Groups and Characters is an effective general textbook on these two fundamentally connected subjects. Presuming only a basic knowledge of abstract algebra as in a first-year graduate course, the text opens with a review of background material and then guides readers carefully through several of the most important aspects of groups and characters, concentrating mainly on finite groups. Challenging yet accessible, Groups and Characters features: * An extensive collection of examples surveying many different types of groups, including Sylow subgroups of symmetric groups, affine groups of fields, the Mathieu groups, and symplectic groups * A thorough, easy-to-follow discussion of Polya-Redfield enumeration, with applications to combinatorics * Inclusive explorations of the transfer function and normal complements, induction and restriction of characters, Clifford theory, characters of symmetric and alternating groups, Frobenius groups, and the Schur index * Illuminating accounts of several computational aspects of group theory, such as the Schreier-Sims algorithm, Todd-Coxeter coset enumeration, and algorithms for generating character tables As valuable as Groups and Characters will prove as a textbook for mathematicians, it has broader applications. With chapters suitable for use as independent review units, along with a full bibliography and index, it will be a dependable general reference for chemists, physicists, and crystallographers.


Book Synopsis Groups and Characters by : Larry C. Grove

Download or read book Groups and Characters written by Larry C. Grove and published by John Wiley & Sons. This book was released on 2011-09-26 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: An authoritative, full-year course on both group theory and ordinary character theory--essential tools for mathematics and the physical sciences One of the few treatments available combining both group theory and character theory, Groups and Characters is an effective general textbook on these two fundamentally connected subjects. Presuming only a basic knowledge of abstract algebra as in a first-year graduate course, the text opens with a review of background material and then guides readers carefully through several of the most important aspects of groups and characters, concentrating mainly on finite groups. Challenging yet accessible, Groups and Characters features: * An extensive collection of examples surveying many different types of groups, including Sylow subgroups of symmetric groups, affine groups of fields, the Mathieu groups, and symplectic groups * A thorough, easy-to-follow discussion of Polya-Redfield enumeration, with applications to combinatorics * Inclusive explorations of the transfer function and normal complements, induction and restriction of characters, Clifford theory, characters of symmetric and alternating groups, Frobenius groups, and the Schur index * Illuminating accounts of several computational aspects of group theory, such as the Schreier-Sims algorithm, Todd-Coxeter coset enumeration, and algorithms for generating character tables As valuable as Groups and Characters will prove as a textbook for mathematicians, it has broader applications. With chapters suitable for use as independent review units, along with a full bibliography and index, it will be a dependable general reference for chemists, physicists, and crystallographers.

The Classical Groups and K-Theory

The Classical Groups and K-Theory

Author: Alexander J. Hahn

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 589

ISBN-13: 3662131528

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It is a great satisfaction for a mathematician to witness the growth and expansion of a theory in which he has taken some part during its early years. When H. Weyl coined the words "classical groups", foremost in his mind were their connections with invariant theory, which his famous book helped to revive. Although his approach in that book was deliberately algebraic, his interest in these groups directly derived from his pioneering study of the special case in which the scalars are real or complex numbers, where for the first time he injected Topology into Lie theory. But ever since the definition of Lie groups, the analogy between simple classical groups over finite fields and simple classical groups over IR or C had been observed, even if the concept of "simplicity" was not quite the same in both cases. With the discovery of the exceptional simple complex Lie algebras by Killing and E. Cartan, it was natural to look for corresponding groups over finite fields, and already around 1900 this was done by Dickson for the exceptional Lie algebras G and E • However, a deep reason for this 2 6 parallelism was missing, and it is only Chevalley who, in 1955 and 1961, discovered that to each complex simple Lie algebra corresponds, by a uniform process, a group scheme (fj over the ring Z of integers, from which, for any field K, could be derived a group (fj(K).


Book Synopsis The Classical Groups and K-Theory by : Alexander J. Hahn

Download or read book The Classical Groups and K-Theory written by Alexander J. Hahn and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 589 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is a great satisfaction for a mathematician to witness the growth and expansion of a theory in which he has taken some part during its early years. When H. Weyl coined the words "classical groups", foremost in his mind were their connections with invariant theory, which his famous book helped to revive. Although his approach in that book was deliberately algebraic, his interest in these groups directly derived from his pioneering study of the special case in which the scalars are real or complex numbers, where for the first time he injected Topology into Lie theory. But ever since the definition of Lie groups, the analogy between simple classical groups over finite fields and simple classical groups over IR or C had been observed, even if the concept of "simplicity" was not quite the same in both cases. With the discovery of the exceptional simple complex Lie algebras by Killing and E. Cartan, it was natural to look for corresponding groups over finite fields, and already around 1900 this was done by Dickson for the exceptional Lie algebras G and E • However, a deep reason for this 2 6 parallelism was missing, and it is only Chevalley who, in 1955 and 1961, discovered that to each complex simple Lie algebra corresponds, by a uniform process, a group scheme (fj over the ring Z of integers, from which, for any field K, could be derived a group (fj(K).

Classical Clifford Algebras

Classical Clifford Algebras

Author: Ilwoo Cho

Publisher: CRC Press

Published: 2024-04-08

Total Pages: 135

ISBN-13: 1040001548

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Classical Clifford Algebras: Operator-Algebraic and Free-Probabilistic Approaches offers novel insights through operator-algebraic and free-probabilistic models. By employing these innovative methods, the author sheds new light on the intrinsic connections between Clifford algebras and various mathematical domains. This monograph should be an essential addition to the library of any researchers interested in Clifford Algebras or Algebraic Geometry more widely. Features Includes multiple examples and applications Suitable for postgraduates and researchers working in Algebraic Geometry Takes an innovative approach to a well-established topic


Book Synopsis Classical Clifford Algebras by : Ilwoo Cho

Download or read book Classical Clifford Algebras written by Ilwoo Cho and published by CRC Press. This book was released on 2024-04-08 with total page 135 pages. Available in PDF, EPUB and Kindle. Book excerpt: Classical Clifford Algebras: Operator-Algebraic and Free-Probabilistic Approaches offers novel insights through operator-algebraic and free-probabilistic models. By employing these innovative methods, the author sheds new light on the intrinsic connections between Clifford algebras and various mathematical domains. This monograph should be an essential addition to the library of any researchers interested in Clifford Algebras or Algebraic Geometry more widely. Features Includes multiple examples and applications Suitable for postgraduates and researchers working in Algebraic Geometry Takes an innovative approach to a well-established topic

Quaternions, Clifford Algebras and Relativistic Physics

Quaternions, Clifford Algebras and Relativistic Physics

Author: Patrick R. Girard

Publisher: Springer Science & Business Media

Published: 2007-06-25

Total Pages: 180

ISBN-13: 3764377917

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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.


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.

Clifford Algebras: An Introduction

Clifford Algebras: An Introduction

Author: D. J. H. Garling

Publisher: Cambridge University Press

Published: 2011-06-23

Total Pages: 210

ISBN-13: 9781107422193

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Clifford algebras, built up from quadratic spaces, have applications in many areas of mathematics, as natural generalizations of complex numbers and the quaternions. They are famously used in proofs of the Atiyah-Singer index theorem, to provide double covers (spin groups) of the classical groups and to generalize the Hilbert transform. They also have their place in physics, setting the scene for Maxwell's equations in electromagnetic theory, for the spin of elementary particles and for the Dirac equation. This straightforward introduction to Clifford algebras makes the necessary algebraic background - including multilinear algebra, quadratic spaces and finite-dimensional real algebras - easily accessible to research students and final-year undergraduates. The author also introduces many applications in mathematics and physics, equipping the reader with Clifford algebras as a working tool in a variety of contexts.


Book Synopsis Clifford Algebras: An Introduction by : D. J. H. Garling

Download or read book Clifford Algebras: An Introduction written by D. J. H. Garling and published by Cambridge University Press. This book was released on 2011-06-23 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt: Clifford algebras, built up from quadratic spaces, have applications in many areas of mathematics, as natural generalizations of complex numbers and the quaternions. They are famously used in proofs of the Atiyah-Singer index theorem, to provide double covers (spin groups) of the classical groups and to generalize the Hilbert transform. They also have their place in physics, setting the scene for Maxwell's equations in electromagnetic theory, for the spin of elementary particles and for the Dirac equation. This straightforward introduction to Clifford algebras makes the necessary algebraic background - including multilinear algebra, quadratic spaces and finite-dimensional real algebras - easily accessible to research students and final-year undergraduates. The author also introduces many applications in mathematics and physics, equipping the reader with Clifford algebras as a working tool in a variety of contexts.

Clifford Algebras

Clifford Algebras

Author: Rafal Ablamowicz

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 635

ISBN-13: 1461220440

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The invited papers in this volume provide a detailed examination of Clifford algebras and their significance to analysis, geometry, mathematical structures, physics, and applications in engineering. While the papers collected in this volume require that the reader possess a solid knowledge of appropriate background material, they lead to the most current research topics. With its wide range of topics, well-established contributors, and excellent references and index, this book will appeal to graduate students and researchers.


Book Synopsis Clifford Algebras by : Rafal Ablamowicz

Download or read book Clifford Algebras written by Rafal Ablamowicz and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 635 pages. Available in PDF, EPUB and Kindle. Book excerpt: The invited papers in this volume provide a detailed examination of Clifford algebras and their significance to analysis, geometry, mathematical structures, physics, and applications in engineering. While the papers collected in this volume require that the reader possess a solid knowledge of appropriate background material, they lead to the most current research topics. With its wide range of topics, well-established contributors, and excellent references and index, this book will appeal to graduate students and researchers.

Clifford Algebras: An Introduction

Clifford Algebras: An Introduction

Author: D. J. H. Garling

Publisher: Cambridge University Press

Published: 2011-06-23

Total Pages: 209

ISBN-13: 1107096383

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A straightforward introduction to Clifford algebras, providing the necessary background material and many applications in mathematics and physics.


Book Synopsis Clifford Algebras: An Introduction by : D. J. H. Garling

Download or read book Clifford Algebras: An Introduction written by D. J. H. Garling and published by Cambridge University Press. This book was released on 2011-06-23 with total page 209 pages. Available in PDF, EPUB and Kindle. Book excerpt: A straightforward introduction to Clifford algebras, providing the necessary background material and many applications in mathematics and physics.

Clifford Algebra to Geometric Calculus

Clifford Algebra to Geometric Calculus

Author: David Hestenes

Publisher: Springer Science & Business Media

Published: 1984

Total Pages: 340

ISBN-13: 9789027725615

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Matrix algebra has been called "the arithmetic of higher mathematics" [Be]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebra' suggested by Clifford himself. Many distinct algebraic systems have been adapted or developed to express geometric relations and describe geometric structures. Especially notable are those algebras which have been used for this purpose in physics, in particular, the system of complex numbers, the quaternions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential forms. Each of these geometric algebras has some significant advantage over the others in certain applications, so no one of them provides an adequate algebraic structure for all purposes of geometry and physics. At the same time, the algebras overlap considerably, so they provide several different mathematical representations for individual geometrical or physical ideas.


Book Synopsis Clifford Algebra to Geometric Calculus by : David Hestenes

Download or read book Clifford Algebra to Geometric Calculus written by David Hestenes and published by Springer Science & Business Media. This book was released on 1984 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt: Matrix algebra has been called "the arithmetic of higher mathematics" [Be]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebra' suggested by Clifford himself. Many distinct algebraic systems have been adapted or developed to express geometric relations and describe geometric structures. Especially notable are those algebras which have been used for this purpose in physics, in particular, the system of complex numbers, the quaternions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential forms. Each of these geometric algebras has some significant advantage over the others in certain applications, so no one of them provides an adequate algebraic structure for all purposes of geometry and physics. At the same time, the algebras overlap considerably, so they provide several different mathematical representations for individual geometrical or physical ideas.