Circular and hyperbolic quaternions, octonions, and sedenions. (English) Zbl 0658.17025
This study contains a survey of the various complex and hypercomplex number algebras up to the 16-dimensional system of the sedenions, introduced by C. Musès [Appl. Math. Comput. 3, 211-226 (1977; Zbl 0359.10050); 4, 45-66, (1978; Zbl 0377.10029) and 6, 63-94 (1980; Zbl 0439.17011)] a system which is alternative, contains nonreal square roots of 1, zero-divisors, nilpotents and a nonreal norm. The author calculates logarithms and polar forms for the algebras in question (quaternions, counterquaternions, octonions, etc.).
Reviewer: A.H.Boers
MSC:
| 17D05 | Alternative rings |
| 17D99 | Other nonassociative rings and algebras |
| 11R52 | Quaternion and other division algebras: arithmetic, zeta functions |
| 11U10 | Nonstandard arithmetic (number-theoretic aspects) |
| 16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |
Keywords:
hypercomplex number algebras; sedenions; nonreal norm; logarithms; polar forms; quaternions; counterquaternions; octonionsReferences:
| [1] | (Halberstam, H.; Ingram, R. E., The Mathematical Papers of Sir William Rowan Hamilton, Vol. III (1967), Cambridge U.P), 103-105 · Zbl 0156.24201 |
| [2] | (Halberstam, H.; Ingram, R. E., The Mathematical Papers of Sir William Rowan Hamilton, Vol. III (1967), Cambridge U.P), 650 · Zbl 0156.24201 |
| [3] | Musès, C., Applied hypernumbers: Computational concepts, Appl. Math. Comput., 3, 211-216 (1976) · Zbl 0359.10050 |
| [4] | Musès, C., Hypernumbers II—Further concepts and computational applications, Appl. Math. Comput., 4, 45-66 (1978) · Zbl 0377.10029 |
| [5] | Musès, C., Hypernumbers and quantum field theory with a summary of physically applicable hypernumber arithmetics and their geometries, Appl. Math. Comput., 6, 63-94 (1980), The modulus of the sedenions (“\(M\)-algebra”) given in this paper is not multiplicative. · Zbl 0439.17011 |
| [6] | Zhevlakov, K. A., Rings That are Nearly Associative, ((1982), Academic), 22-34 · Zbl 0487.17001 |
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