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Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups
Universitext
ISBN/EAN: 9780387941103
Umbreit-Nr.: 4149251
Sprache:
Englisch
Umfang: xi, 286 S.
Format in cm:
Einband:
kartoniertes Buch
Erschienen am 17.12.1993
Auflage: 1/1994
- Zusatztext
- InhaltsangabeNotation 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.