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Author: Alexander J. Hahn Publisher: Springer Science & Business Media ISBN: 146846311X Category : Mathematics Languages : en Pages : 296
Book Description
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: Alexander J. Hahn Publisher: Springer Science & Business Media ISBN: 146846311X Category : Mathematics Languages : en Pages : 296
Book Description
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: Goro Shimura Publisher: American Mathematical Soc. ISBN: 1470415623 Category : Education Languages : en Pages : 290
Book Description
In this book, award-winning author Goro Shimura treats new areas and presents relevant expository material in a clear and readable style. Topics include Witt's theorem and the Hasse principle on quadratic forms, algebraic theory of Clifford algebras, spin groups, and spin representations. He also includes some basic results not readily found elsewhere. The two principle themes are: (1) Quadratic Diophantine equations; (2) Euler products and Eisenstein series on orthogonal groups and Clifford groups. The starting point of the first theme is the result of Gauss that the number of primitive representations of an integer as the sum of three squares is essentially the class number of primitive binary quadratic forms. Presented are a generalization of this fact for arbitrary quadratic forms over algebraic number fields and various applications. For the second theme, the author proves the existence of the meromorphic continuation of a Euler product associated with a Hecke eigenform on a Clifford or an orthogonal group. The same is done for an Eisenstein series on such a group. Beyond familiarity with algebraic number theory, the book is mostly self-contained. Several standard facts are stated with references for detailed proofs. Goro Shimura won the 1996 Steele Prize for Lifetime Achievement for "his important and extensive work on arithmetical geometry and automorphic forms".
Author: W. Scharlau Publisher: Springer Science & Business Media ISBN: 3642699715 Category : Mathematics Languages : en Pages : 431
Book Description
For a long time - at least from Fermat to Minkowski - the theory of quadratic forms was a part of number theory. Much of the best work of the great number theorists of the eighteenth and nineteenth century was concerned with problems about quadratic forms. On the basis of their work, Minkowski, Siegel, Hasse, Eichler and many others crea ted the impressive "arithmetic" theory of quadratic forms, which has been the object of the well-known books by Bachmann (1898/1923), Eichler (1952), and O'Meara (1963). Parallel to this development the ideas of abstract algebra and abstract linear algebra introduced by Dedekind, Frobenius, E. Noether and Artin led to today's structural mathematics with its emphasis on classification problems and general structure theorems. On the basis of both - the number theory of quadratic forms and the ideas of modern algebra - Witt opened, in 1937, a new chapter in the theory of quadratic forms. His most fruitful idea was to consider not single "individual" quadratic forms but rather the entity of all forms over a fixed ground field and to construct from this an algebra ic object. This object - the Witt ring - then became the principal object of the entire theory. Thirty years later Pfister demonstrated the significance of this approach by his celebrated structure theorems.
Author: Eckhard Meinrenken Publisher: Springer Science & Business Media ISBN: 3642362168 Category : Mathematics Languages : en Pages : 331
Book Description
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: Krishnaswami Alladi Publisher: Springer Science & Business Media ISBN: 1461474884 Category : Mathematics Languages : en Pages : 303
Book Description
In the last decade, the areas of quadratic and higher degree forms have witnessed dramatic advances. This volume is an outgrowth of three seminal conferences on these topics held in 2009, two at the University of Florida and one at the Arizona Winter School. The volume also includes papers from the two focused weeks on quadratic forms and integral lattices at the University of Florida in 2010.Topics discussed include the links between quadratic forms and automorphic forms, representation of integers and forms by quadratic forms, connections between quadratic forms and lattices, and algorithms for quaternion algebras and quadratic forms. The book will be of interest to graduate students and mathematicians wishing to study quadratic and higher degree forms, as well as to established researchers in these areas. Quadratic and Higher Degree Forms contains research and semi-expository papers that stem from the presentations at conferences at the University of Florida as well as survey lectures on quadratic forms based on the instructional workshop for graduate students held at the Arizona Winter School. The survey papers in the volume provide an excellent introduction to various aspects of the theory of quadratic forms starting from the basic concepts and provide a glimpse of some of the exciting questions currently being investigated. The research and expository papers present the latest advances on quadratic and higher degree forms and their connections with various branches of mathematics.
Author: Jacques Helmstetter Publisher: Springer Science & Business Media ISBN: 3764386061 Category : Mathematics Languages : en Pages : 512
Book Description
After general properties of quadratic mappings over rings, the authors more intensely study quadratic forms, and especially their Clifford algebras. To this purpose they review the required part of commutative algebra, and they present a significant part of the theory of graded Azumaya algebras. Interior multiplications and deformations of Clifford algebras are treated with the most efficient methods.
Author: McMaster University Publisher: American Mathematical Soc. ISBN: 9780821860083 Category : Mathematics Languages : en Pages : 362
Book Description
Contains the proceedings of the 1983 Seminar on Quadratic and Hermitian Forms held at McMaster University, July 1983. Between 1945 and 1965, most of the work in quadratic (and hermitian) forms took place in arithmetic theory (M Eichler, M Kneser, O T O'Meara).
Author: Pertti Lounesto Publisher: Springer Science & Business Media ISBN: 9780817635251 Category : Mathematics Languages : en Pages : 664
Book Description
In addition, attention is paid to the algebraic and Lie-theoretic applications of Clifford algebras---particularly their intersection with Hopf algebras, Lie algebras and representations, graded algebras, and associated mathematical structures. Symplectic Clifford algebras are also discussed. Finally, Clifford algebras play a strong role in both physics and engineering. The physics section features an investigation of geometric algebras, chiral Dirac equations, spinors and Fermions, and applications of Clifford algebras in classical mechanics and general relativity. Twistor and octonionic methods, electromagnetism and gravity, elementary particle physics, noncommutative physics, Dirac's equation, quantum spheres, and the Standard Model are among topics considered at length.
Author: Jean-Luc Brylinski Publisher: Springer Science & Business Media ISBN: 1461217709 Category : Mathematics Languages : en Pages : 404
Book Description
This book is an outgrowth of the activities of the Center for Geometry and Mathematical Physics (CGMP) at Penn State from 1996 to 1998. The Center was created in the Mathematics Department at Penn State in the fall of 1996 for the purpose of promoting and supporting the activities of researchers and students in and around geometry and physics at the university. The CGMP brings many visitors to Penn State and has ties with other research groups; it organizes weekly seminars as well as annual workshops The book contains 17 contributed articles on current research topics in a variety of fields: symplectic geometry, quantization, quantum groups, algebraic geometry, algebraic groups and invariant theory, and character istic classes. Most of the 20 authors have talked at Penn State about their research. Their articles present new results or discuss interesting perspec tives on recent work. All the articles have been refereed in the regular fashion of excellent scientific journals. Symplectic geometry, quantization and quantum groups is one main theme of the book. Several authors study deformation quantization. As tashkevich generalizes Karabegov's deformation quantization of Kahler manifolds to symplectic manifolds admitting two transverse polarizations, and studies the moment map in the case of semisimple coadjoint orbits. Bieliavsky constructs an explicit star-product on holonomy reducible sym metric coadjoint orbits of a simple Lie group, and he shows how to con struct a star-representation which has interesting holomorphic properties.
Author: Alexander J. Hahn
Author: Alexander J. Hahn
Author: Alexander J Hahn
Author: Goro Shimura
Author: W. Scharlau
Author: Eckhard Meinrenken
Author: Krishnaswami Alladi
Author: Jacques Helmstetter
Author: McMaster University
Author: Pertti Lounesto
Author: Jean-Luc Brylinski