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Hypercomplex Numbers - An Elementary Introduction to Algebras
Posted on 2010-03-16
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More I.L. Kantor & A.S. Solodovnikov "" The problem of finding all hypercomplex number system remains unsolved. The authors consider some variants of this problem. They begin with examples including the quaternions and Cayley numbers; the second (auxiliary) part is an elementary exposition of linear algebra; the third part explains the unique role of quaternions, complex- and Cayley numbers in the problem. Neither index nor bibliography. Table of Contents: I. Hypercomplex Numbers 1. Complex Numbers 2. Alternate Arithmetics on the Numbers a + bi 3. Quaternions 4. Quaternions and Vector Algebra 5. Hypercomplex Numbers 6. The Doubling Procedure. Cayley Numbers 7. Algebras II. N-Dimensional Vectors 8. The N-Dimensional Vector Space An 9. A Basis of the Space An 10. Subspaces 11. Lemma on Homogeneous Systems of Equations 12. Scalar Products 13. Orthonormal Basis. Orthogonal Transformation III. The Exceptional Position of Four Algebras 14. Isomorphic Algebra 15. Subalgebras 16. Translation of the "Problem of the Sum of Squares" into the Language of Algebras. Normed Algebras 17. Normed Algebras with an Identity. Hurwitz's Theorem 18. A Method for Constructing All Normed Algebras and Its Implications for the Problm of the Sum of Squares 19. Frobenius' Theorem 20. Commutative Division Algebras 21. Conclusion 22. Notes Please post mirrors.
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