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arXiv · 2606.27798

Involutions in the Cayley-Dickson construction

Masood Aryapoor·Per Bäck·Sophie Pautrel
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Abstract

We determine all involutions in the Cayley-Dickson construction that extend the involution of the original $*$-algebra. We also find all algebra isomorphisms between the resulting Cayley doubles that extend the identity automorphism of the original $*$-algebra, and consequently classify the resulting $*$-algebras up to $*$-algebra isomorphism. As applications, we show that Cayley doubles without zero divisors admit exactly one additional involution, prove that the classical Cayley-Dickson involution is the unique scalar involution, and obtain a classification of the $*$-algebras arising from $\mathbb{R}$ up to dimension 4.

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Masood Aryapoor, Per Bäck, Sophie Pautrel. 2026-06-26. Involutions in the Cayley-Dickson construction. https://arxiv.org/abs/2606.27798

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