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

TL;DR

This paper classifies Rota-Baxter operators of nonzero weight on split octonions, identifying a unique non-splitting operator and describing algebra decompositions, thereby completing the classification on composition algebras.

Contribution

It provides a complete classification of Rota-Baxter operators on split octonions and describes algebra decompositions over certain fields, extending previous work on composition algebras.

Findings

Exactly one non-splitting Rota-Baxter operator exists up to transformations.

All decompositions of split octonions into subalgebras are characterized.

Classification of Rota-Baxter operators on composition algebras is completed.

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.