RETRACTED: An alternate Cayley-Dickson product. (English) Zbl 1410.17004

Bales, John W.

Missouri J. Math. Sci. 28, No. 1, 88-96 (2016); retraction note ibid. 32, No.1, 118 (2020).

Summary: Although the Cayley-Dickson algebras are twisted group algebras, little attention has been paid to the nature of the Cayley-Dickson twist. One reason is that the twist appears to be highly chaotic and there are other interesting things about the algebras to focus attention upon. However, if one uses a doubling product for the algebras different from yet equivalent to the ones commonly used, and if one uses a numbering of the basis vectors different from the standard basis a quite beautiful and highly periodic twist emerges. This leads easily to a simple closed form equation for the product of any two basis vectors of a Cayley-Dickson algebra.
[This article has been retracted by the author. He has discovered that the doubling product, which is the primary subject of the paper, does not generate a Cayley-Dickson algebra beyond the second doubling. As the paper cannot be repaired, he has asked for the paper’s retraction.]

MSC:

17A60

Structure theory for nonassociative algebras

Keywords:

Cayley-Dickson; doubling product; twisted group product; fractal; twist tree