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.

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 ofS.- 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.