正交和辛克利福德代数 Orthogonal and Symplectic Clifford Algebras 英文原版 A Crumeyrolle

基本信息

Format Paperback | 350 pages

Dimensions 155 x 235 x 19.3mm | 563g

Publication date 09 Dec 2010

Publisher Springer

Language English

Edition Statement Softcover reprint of hardcover 1st ed. 1989

Illustrations note XIV, 350 p.

ISBN10 9048140595

ISBN13 9789048140596

书籍简介

这卷提出了克利福德代数和旋量的理论的系统发展，和他们在微分几何和数学物理的主要应用。在克利福德代数中，旋量通常作为小左理想的元素出现。通常的算子(共轭、推导)被定义为所有的特征和所有的维数;扭体以旋量的形式出现，广义试态原理导致了超对称性。关于脊柱性的新概念。介绍了群旋量结构、放大旋量结构和非晶旋量结构。以一种平行的方式，辛克利福德代数的定义-给出了一个有用的推广的Weyl代数-连同棘旋群和辛旋量。这允许几何方法接近马斯洛夫指数和构造代数变形的任意辛流形。在附录中给出了幂等束的研究、可微情况下所谓的彭罗斯变换的研究以及辛旋量场上傅里叶变换的几何解释。

This volume presents a systematic development of the theory of Clifford algebras and spinors, and their principal applications to differential geometry and mathematical physics. Most often, spinors occur as elements of minimal left ideals in the Clifford algebra. A constructive, geometric approach is used, especially in the theory of spinor bundles, where the local frames are mentioned explicitly The usual operators (conjugations, derivations) are defined for all signatures and all dimensions; twistors appear as spinors and the generalized triality principle leads to supersymmetries. New notions of the spinoriality . group, enlarged spinor structure and amorphic spinor structure are introduced. In a parallel way, symplectic Clifford algebras are defined -giving a useful generalization of the Weyl algebras - along with the spinoplectic groups and the symplectic spinors. This allows a geometric approach to the Maslov index and to the construction of algebra deformations over arbitrary symplectic manifolds. The study of idempotent bundles, that of the so-called Penrose transform in the differentiable case and a geometric interpretation of the Fourier transform on the symplectic spinor fields are given in appendices. 

目录

Orthogonal and Symplectic Geometries.- Tensor Algebras, Exterior Algebras and Symmetric Algebras.- Orthogonal Clifford Algebras.- The Clifford Groups, the Twisted Clifford Groups and Their Fundamental Subgroups.- Spinors and Spin Representations.- Fundamental Lie Algebras and Lie Groups in the Clifford Algebras.- The Matrix Approach to Spinors in Three and Four-Dimensional Spaces.- The Spinors in Maximal Index and Even Dimension.- The Spinors in Maximal Index and Odd Dimension.- The Hermitian Structure on the Space of Complex Spinors-Conjugations and Related Notions.- Spinoriality Groups.- Coverings of the Complete Conformal Group-Twistors.- The Triality Principle, the Interaction Principle and Orthosymplectic Graded Lie Algebras.- The Clifford Algebra and the Clifford Bundle of a Pseudo-Riemannian Manifold. Existence Conditions for Spinor Structures.- Spin Derivations.- The Dirac Equation.- Symplectic Clifford Algebras and Associated Groups.- Symplectic Spinor Bundles-The Maslov Index.- Algebra Deformations on Symplectic Manifolds.- The Primitive Idempotents of the Clifford Algebras and the Amorphic Spinor Fiber Bundles.- Self-Dual Yang-Mills Fields and the Penrose Transform in the Spinor Context.- Symplectic Structures, Complex, Structures, Symplectic Spinors and the Fourier Transform.

作者简介

Albert J. Crumeyrolle (1919–1992) 是法国数学家和保罗萨巴蒂尔大学数学教授，以其对旋量结构和克利福德代数的贡献而闻名。

Albert J. Crumeyrolle (1919–1992) was a French mathematician and professor of mathematics at the Paul Sabatier University, known for his contributions to spinor structures and Clifford algebra.