Rota-Baxter operators of nonzero weight on the split Cayley-Dickson algebra

TL;DR

This paper fully classifies Rota-Baxter operators of nonzero weight on the split Cayley-Dickson algebra $\mathbb{O}$ by proving that non-splitting operators are, up to automorphisms, anti-automorphisms, scalar rescalings, and the $\phi$-transformation, essentially unique and equivalent to a canonical operator $R_1$ with explicit action on subspaces. It shows that nonzero-weight non-splitting operators have kernel dimension $3$ or $4$, constructing canonical representatives $R_1$ (kernel $4$) and $R_2$ (kernel $3$) and establishing their equivalence under the allowed symmetries. It then classifies all decompositions of split octonions into two subalgebras over a quadratically closed field of characteristic not $2$, enumerating seven nonisomorphic cases that correspond to splitting Rota-Baxter operators. These results complete the classification of Rota-Baxter operators on split composition algebras for any weight and connect operator theory with the algebraic structure of split octonions via subalgebra decompositions.

Abstract

We describe Rota-Baxter operators on split octonions. It turns out that up to some transformations there exists exactly one such non-splitting operator over any field. We also obtain a description of all decompositions of split octonions over a quadratically closed field of characteristic different from 2 into a sum of two subalgebras, which describes the splitting Rota-Baxter operators. It completes the classification of Rota-Baxter operators on composition algebras of any weight.