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Author: Tonny A. Springer
Publisher: Springer
Published: 2013-12-21
Total Pages: 212
ISBN-13: 3662126222
DOWNLOAD EBOOKThe 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.
Author: Tonny A. Springer
Publisher:
Published: 2014-09-01
Total Pages: 220
ISBN-13: 9783662126233
DOWNLOAD EBOOKAuthor: N. Jacobson
Publisher: Routledge
Published: 2017-10-19
Total Pages: 140
ISBN-13: 1351449389
DOWNLOAD EBOOKThis volume presents a set of models for the exceptional Lie algebras over algebraically closed fieldsof characteristic O and over the field of real numbers. The models given are based on the algebras ofCayley numbers (octonions) and on exceptional Jordan algebras. They are also valid forcharacteristics p * 2. The book also provides an introduction to the problem of forms of exceptionalsimple Lie algebras, especially the exceptional D4 's, 6 's, and 7 's. These are studied by means ofconcrete realizations of the automorphism groups.Exceptional Lie Algebras is a useful tool for the mathematical public in general-especially thoseinterested in the classification of Lie algebras or groups-and for theoretical physicists.
Author: John H. Conway
Publisher: CRC Press
Published: 2003-01-23
Total Pages: 172
ISBN-13: 1439864187
DOWNLOAD EBOOKThis 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
Author: Tevian Dray
Publisher: World Scientific
Published: 2015-04-08
Total Pages: 229
ISBN-13: 9814401838
DOWNLOAD EBOOKThere 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.
Author: John R. Faulkner
Publisher:
Published: 1970
Total Pages: 71
ISBN-13: 9780821818046
DOWNLOAD EBOOKMany of the results originally due to Springer, Veldkamp, Jacobson, Suh, and others on octonion planes are derived here in a uniform fashion, valid for all characteristics and for both split and division octonion algebras, by using an exceptional quadratic Jordan algebra. Among the subjects treated are norm semi-similarities of an exceptional quadratic Jordan algebra, an isomorphism of a spin group with a subgroup of norm preserving maps, the "fundamental theorem" for octonion planes, the harmonic point theorem, simplicity and isomorphisms of the "little projective group," automorphisms of order two in an octonion algebra in characteristic two, unitary groups of collineations commuting with a polarity, and the simplicity of the automorphism group of the exceptional quadratic Jordan algebra in characteristic two.
Author: Predrag Cvitanović
Publisher: Princeton University Press
Published: 2008-07-01
Total Pages: 288
ISBN-13: 9781400837670
DOWNLOAD EBOOKIf classical Lie groups preserve bilinear vector norms, what Lie groups preserve trilinear, quadrilinear, and higher order invariants? Answering this question from a fresh and original perspective, Predrag Cvitanovic takes the reader on the amazing, four-thousand-diagram journey through the theory of Lie groups. This book is the first to systematically develop, explain, and apply diagrammatic projection operators to construct all semi-simple Lie algebras, both classical and exceptional. The invariant tensors are presented in a somewhat unconventional, but in recent years widely used, "birdtracks" notation inspired by the Feynman diagrams of quantum field theory. Notably, invariant tensor diagrams replace algebraic reasoning in carrying out all group-theoretic computations. The diagrammatic approach is particularly effective in evaluating complicated coefficients and group weights, and revealing symmetries hidden by conventional algebraic or index notations. The book covers most topics needed in applications from this new perspective: permutations, Young projection operators, spinorial representations, Casimir operators, and Dynkin indices. Beyond this well-traveled territory, more exotic vistas open up, such as "negative dimensional" relations between various groups and their representations. The most intriguing result of classifying primitive invariants is the emergence of all exceptional Lie groups in a single family, and the attendant pattern of exceptional and classical Lie groups, the so-called Magic Triangle. Written in a lively and personable style, the book is aimed at researchers and graduate students in theoretical physics and mathematics.
Author: Feza Grsey
Publisher: World Scientific
Published: 1996
Total Pages: 492
ISBN-13: 9789810228637
DOWNLOAD EBOOKThis monograph surveys the role of some associative and non-associative algebras, remarkable by their ubiquitous appearance in contemporary theoretical physics, particularly in particle physics. It concerns the interplay between division algebras, specifically quaternions and octonions, between Jordan and related algebras on the one hand, and unified theories of the basic interactions on the other. Selected applications of these algebraic structures are discussed: quaternion analyticity of Yang-Mills instantons, octonionic aspects of exceptional broken gauge, supergravity theories, division algebras in anyonic phenomena and in theories of extended objects in critical dimensions. The topics presented deal primarily with original contributions by the authors.
Author: Kevin McCrimmon
Publisher: Springer Science & Business Media
Published: 2006-05-29
Total Pages: 584
ISBN-13: 0387217967
DOWNLOAD EBOOKThis 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.
Author: B. Rosenfeld
Publisher: Springer Science & Business Media
Published: 2013-03-09
Total Pages: 414
ISBN-13: 147575325X
DOWNLOAD EBOOKThis book is the result of many years of research in Non-Euclidean Geometries and Geometry of Lie groups, as well as teaching at Moscow State University (1947- 1949), Azerbaijan State University (Baku) (1950-1955), Kolomna Pedagogical Col lege (1955-1970), Moscow Pedagogical University (1971-1990), and Pennsylvania State University (1990-1995). My first books on Non-Euclidean Geometries and Geometry of Lie groups were written in Russian and published in Moscow: Non-Euclidean Geometries (1955) [Ro1] , Multidimensional Spaces (1966) [Ro2] , and Non-Euclidean Spaces (1969) [Ro3]. In [Ro1] I considered non-Euclidean geometries in the broad sense, as geometry of simple Lie groups, since classical non-Euclidean geometries, hyperbolic and elliptic, are geometries of simple Lie groups of classes Bn and D , and geometries of complex n and quaternionic Hermitian elliptic and hyperbolic spaces are geometries of simple Lie groups of classes An and en. [Ro1] contains an exposition of the geometry of classical real non-Euclidean spaces and their interpretations as hyperspheres with identified antipodal points in Euclidean or pseudo-Euclidean spaces, and in projective and conformal spaces. Numerous interpretations of various spaces different from our usual space allow us, like stereoscopic vision, to see many traits of these spaces absent in the usual space.
