Clifford Algebras And Lie Theory PDF Download
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Author | : Eckhard Meinrenken |
Publisher | : Springer Science & Business Media |
Total Pages | : 331 |
Release | : 2013-02-28 |
Genre | : Mathematics |
ISBN | : 3642362168 |
Download Clifford Algebras and Lie Theory Book in PDF, ePub and Kindle
This monograph provides an introduction to the theory of Clifford algebras, with an emphasis on its connections with the theory of Lie groups and Lie algebras. The book starts with a detailed presentation of the main results on symmetric bilinear forms and Clifford algebras. It develops the spin groups and the spin representation, culminating in Cartan’s famous triality automorphism for the group Spin(8). The discussion of enveloping algebras includes a presentation of Petracci’s proof of the Poincaré–Birkhoff–Witt theorem. This is followed by discussions of Weil algebras, Chern--Weil theory, the quantum Weil algebra, and the cubic Dirac operator. The applications to Lie theory include Duflo’s theorem for the case of quadratic Lie algebras, multiplets of representations, and Dirac induction. The last part of the book is an account of Kostant’s structure theory of the Clifford algebra over a semisimple Lie algebra. It describes his “Clifford algebra analogue” of the Hopf–Koszul–Samelson theorem, and explains his fascinating conjecture relating the Harish-Chandra projection for Clifford algebras to the principal sl(2) subalgebra. Aside from these beautiful applications, the book will serve as a convenient and up-to-date reference for background material from Clifford theory, relevant for students and researchers in mathematics and physics.
Author | : Alexander J. Hahn |
Publisher | : Springer Science & Business Media |
Total Pages | : 296 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 146846311X |
Download Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups Book in PDF, ePub and Kindle
Quadratic Algebras, Clifford Algebras, and Arithmetic Forms introduces mathematicians to the large and dynamic area of algebras and forms over commutative rings. The book begins very elementary and progresses gradually in its degree of difficulty. Topics include the connection between quadratic algebras, Clifford algebras and quadratic forms, Brauer groups, the matrix theory of Clifford algebras over fields, Witt groups of quadratic and symmetric bilinear forms. Some of the new results included by the author concern the representation of Clifford algebras, the structure of Arf algebra in the free case, connections between the group of isomorphic classes of finitely generated projectives of rank one and arithmetic results about the quadratic Witt group.
Author | : Ian R. Porteous |
Publisher | : Cambridge University Press |
Total Pages | : 309 |
Release | : 1995-10-05 |
Genre | : Mathematics |
ISBN | : 0521551773 |
Download Clifford Algebras and the Classical Groups Book in PDF, ePub and Kindle
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.
Author | : David Hestenes |
Publisher | : Springer Science & Business Media |
Total Pages | : 340 |
Release | : 1984 |
Genre | : Mathematics |
ISBN | : 9789027725615 |
Download Clifford Algebra to Geometric Calculus Book in PDF, ePub and Kindle
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.
Author | : Rafal Ablamowicz |
Publisher | : Springer Science & Business Media |
Total Pages | : 635 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1461220440 |
Download Clifford Algebras Book in PDF, ePub and Kindle
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.
Author | : George B. Seligman |
Publisher | : Springer |
Total Pages | : 203 |
Release | : 2006-11-14 |
Genre | : Mathematics |
ISBN | : 3540388648 |
Download Constructions of Lie Algebras and their Modules Book in PDF, ePub and Kindle
This book deals with central simple Lie algebras over arbitrary fields of characteristic zero. It aims to give constructions of the algebras and their finite-dimensional modules in terms that are rational with respect to the given ground field. All isotropic algebras with non-reduced relative root systems are treated, along with classical anisotropic algebras. The latter are treated by what seems to be a novel device, namely by studying certain modules for isotropic classical algebras in which they are embedded. In this development, symmetric powers of central simple associative algebras, along with generalized even Clifford algebras of involutorial algebras, play central roles. Considerable attention is given to exceptional algebras. The pace is that of a rather expansive research monograph. The reader who has at hand a standard introductory text on Lie algebras, such as Jacobson or Humphreys, should be in a position to understand the results. More technical matters arise in some of the detailed arguments. The book is intended for researchers and students of algebraic Lie theory, as well as for other researchers who are seeking explicit realizations of algebras or modules. It will probably be more useful as a resource to be dipped into, than as a text to be worked straight through.
Author | : John E. Gilbert |
Publisher | : Cambridge University Press |
Total Pages | : 346 |
Release | : 1991-07-26 |
Genre | : Mathematics |
ISBN | : 9780521346542 |
Download Clifford Algebras and Dirac Operators in Harmonic Analysis Book in PDF, ePub and Kindle
The aim of this book is to unite the seemingly disparate topics of Clifford algebras, analysis on manifolds, and harmonic analysis. The authors show how algebra, geometry, and differential equations play a more fundamental role in Euclidean Fourier analysis. They then link their presentation of the Euclidean theory naturally to the representation theory of semi-simple Lie groups.
Author | : Patrick R. Girard |
Publisher | : Springer Science & Business Media |
Total Pages | : 177 |
Release | : 2007-06-25 |
Genre | : Mathematics |
ISBN | : 3764377917 |
Download Quaternions, Clifford Algebras and Relativistic Physics Book in PDF, ePub and Kindle
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.
Author | : Hossein Abbaspour |
Publisher | : World Scientific |
Total Pages | : 444 |
Release | : 2007 |
Genre | : Science |
ISBN | : 9812706984 |
Download Basic Lie Theory Book in PDF, ePub and Kindle
This volume provides a comprehensive treatment of basic Lie theory, primarily directed toward graduate study. The text is ideal for a full graduate course in Lie groups and Lie algebras. However, the book is also very usable for a variety of other courses: a one-semester course in Lie algebras, or on Haar measure and its applications, for advanced undergraduates; or as the text for one-semester graduate courses in Lie groups and symmetric spaces of non-compact type, or in lattices in Lie groups. The material is complete and detailed enough to be used for self-study; it can also serve as a reference work for professional mathematicians working in other areas. The book's utility for such a varied readership is enhanced by a diagram showing the interdependence of the separate chapters so that individual chapters and the material they depend upon can be selected, while others can be skipped.The book incorporates many of the most significant discoveries and pioneering contributions of the masters of the subject: Borel, Cartan, Chevalley, Iwasawa, Mostow, Siegel, and Weyl, among others.
Author | : Adam M. Bincer |
Publisher | : Oxford University Press |
Total Pages | : 216 |
Release | : 2013 |
Genre | : Mathematics |
ISBN | : 0199662924 |
Download Lie Groups and Lie Algebras - A Physicist's Perspective Book in PDF, ePub and Kindle
This book is intended for graduate students in Physics. It starts with a discussion of angular momentum and rotations in terms of the orthogonal group in three dimensions and the unitary group in two dimensions and goes on to deal with these groups in any dimensions. All representations of su(2) are obtained and the Wigner-Eckart theorem is discussed. Casimir operators for the orthogonal and unitary groups are discussed. The exceptional group G2 is introduced as the group of automorphisms of octonions. The symmetric group is used to deal with representations of the unitary groups and the reduction of their Kronecker products. Following the presentation of Cartan's classification of semisimple algebras Dynkin diagrams are described. The book concludes with space-time groups - the Lorentz, Poincare and Liouville groups - and a derivation of the energy levels of the non-relativistic hydrogen atom in n space dimensions.