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Large annihilators in Cayley-Dickson algebras

Daniel K. Biss·Daniel Dugger·Daniel C. Isaksen
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Abstract

Cayley-Dickson algebras are an infinite sequence of non-associative algebras starting with the reals, complexes, quaternions, and octonions. We study the zero-divisors in the higher Cayley-Dickson algebras. In particular, we show that the annihilator of any element in the 2^n-dimensional Cayley-Dickson algebra has dimension at most 2^n-4n+4. Moroever, every multiple of four between 0 and this upper bound actually occurs as the dimension of some annihilator (a theorem of Moreno says that only multiples of four can occur). We completely describe all the zero-divisors whose annihilator has dimension 2^n-4n+4.

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Daniel K. Biss, Daniel Dugger, Daniel C. Isaksen. 2005-11-28. Large annihilators in Cayley-Dickson algebras. https://arxiv.org/abs/math/0511691

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