Quadratic Algebras Clifford Algebras And Arithmetic Witt Groups

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.

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.

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 st

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.

The two main themes of the book are (1) quadratic Diophantine equations; (2) Euler products and Eisenstein series on orthogonal groups and Clifford groups. Whereas the latest chapters of the book contain new results, a substantial portion of it is devoted to expository material related to these themes, such as Witt's theorem and the Hasse principle on quadratic forms, algebraic theory of Clifford algebras, spin groups, and spin representations. The starting point of the first main 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. A generalization of this fact for arbitrary quadratic forms over algebraic number fields, as well as various applications are presented. As for the second theme, the existence of the meromorphic continuation of an Euler product associated with a Hecke eigenform on a Clifford or an orthogonal group is proved. The same is done for an Eisenstein series on such a group. The book is practically self-contained, except that familiarity with algebraic number theory is assumed and several standard facts are stated without detailed proof, but with precise references.

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.

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.