Author | Hahn, Alexander J. author |
---|---|
Title | Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups [electronic resource] / by Alexander J. Hahn |
Imprint | New York, NY : Springer US, 1994 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-6311-8 |
Descript | XI, 286p. 18 illus. online resource |
Notation and Terminology -- 1. Fundamental Concepts in the Theory of Algebras -- A. Free Quadratic Algebras -- B. Involutions on Algebras -- C. Gradings on Algebras -- D. Tensor Products and Graded Tensor Products -- E. Exercises -- 2. Separable Algebras -- A. Separability of Algebras -- B. Separability Idempotents -- C. Separable Free Quadratic Algebras -- D. Properties of Conjugation -- E. Exercises -- 3. Groups of Free Quadratic Algebras -- A. The Group Quf(R) -- B. The Discriminant ? -- C. The Group QUf(R) -- D. Another Look at (a, b)? * (b, c)? -- E. Exercises -- 4. Bilinear and Quadratic Forms -- A. Localization -- B. Bilinear Forms -- C. The Group Dis(R) -- D. Quadratic Forms -- E. Exercises -- 5. Clifford Algebras: The Basics -- A. Definition and Existence -- B. Generation, Grading, and Involutions -- C. Graded Tensor Product -- D. Exterior Algebras -- E. Exercises -- 6. Algebras with Standard Involution -- A. Standard Involutions -- B. Free Quaternion Algebras -- C. Separability of Free Quaternion Algebras -- D. Nonsingular Algebras -- E. Exercises -- 7. Arf Algebras and Special Elements -- A. TheArf Algebra -- B. The Arf Algebra of an Orthogonal Sum -- C. Special Elements -- D. Exercises -- 8. Consequences of the Existence of Special Elements -- A. Connections between C(M) and C0(M) -- B. Gradings Defined by Roots of X2 - aX - b -- C. Linear Maps with Polynomial X2 - aX - b -- D. Graded Properties of Representations -- E. Comparing the Tensor and Graded Tensor Products -- F. Exercises -- 9. Structure of Clifford and Arf Algebras -- A. More on Separable Algebras -- B. The Separability of C(M) and C0(M) -- C. The Even-Odd Splitting of C(M) -- D. The Structures of Cen C(M), Cen C0(M), and A(M) -- E. Exercises -- 10. The Existence of Special Elements -- A. Separable Quadratic Algebras -- B. The Discriminant Module of S -- C. Criteria for the Existence of Special Elements -- D. Special Elements and the Discriminant -- E. Exercises -- 11. Matrix Theory of Clifford Algebras over Fields -- A. Matrix Connections between C(M) and C0(M) -- B. Basics about Quadratic Spaces -- C. Quaternion Algebras -- D. Periodicity Phenomena -- E. Local and Global Fields -- F. Exercises -- 12. Dis(R) and Qu(R) -- A. The Quadratic Group Qu(R) -- B. More about Dis(R) -- C. Connecting Qu(R) with Dis(R) -- D. The Case of an Integrally Closed Domain -- E. The Classical Discriminant -- F. Exercises -- 13. Brauer Groups and Witt Groups -- A. Brauer and Brauer-Wall Groups -- B. The Graded Quadratic Group QU(R) -- C. The Witt Group of Quadratic Forms -- D. The Witt Group of Symmetric Bilinear Forms -- E. The Classical Situations -- F. Exercises -- 14. The Arithmetic of Wq(R) -- A. Arithmetic Dedekind Domains -- B. The Arithmetic of Br(R)2 -- C. AnalyzingWq(R) -- D. Computing Qu(R?) and Wq(R?) -- E. Connections between W(R) and Wq(R) -- F. Exercises -- 15. Applications of Clifford Modules -- A. Clifford Modules -- B. Vector Fields on Spheres -- C. Connections with Topological K-Theory -- D. Lie Groups and Lie Algebras -- E. Dirac Operators -- F. Spin Manifolds -- G. Isoparametric Hypersurfaces