Abstract

The Cayley-Dickson algebras A_n are an infinite sequence of (non-associative) algebras beginning with the well-known composition algebras ℝ, ℂ, ℍ, 𝕆. We completely describe all possible dimensions for the alternator Alt(a) ≔ {b ∈ A_n | a(ab) = (aa)b = 0} of an element a ∈ A_n, for n ≥ 7. This resolves a conjecture of Biss, Christensen, Dugger, and Isaksen. On the way to obtaining this result, we establish numerous results on the eigentheory of left multiplication operators in A_n, some of which may be of independent interest.