Rota-Baxter Operators on Quadratic Algebras

TL;DR

This paper classifies Rota-Baxter operators on various algebraic structures, revealing their triviality or specific forms, and connects them to solutions of the Yang-Baxter equation.

Contribution

It provides a comprehensive analysis of Rota-Baxter operators on quadratic algebras, including classification results and links to the Yang-Baxter equation.

Findings

All Rota-Baxter operators on quadratic division algebras are trivial.

On simple odd-dimensional Jordan algebras, operators are projections on subalgebras.

Connections established between Rota-Baxter operators and the alternative Yang-Baxter equation.

Abstract

We prove that all Rota-Baxter operators on a quadratic division algebra are trivial. For nonzero weight, we state that all Rota-Baxter operators on the simple odd-dimensional Jordan algebra of bilinear form are projections on a subalgebra along another one. For weight zero, we find a connection between the Rota-Baxter operators and the solutions to the alternative Yang-Baxter equation on the Cayley-Dickson algebra. We also investigate the Rota-Baxter operators on the matrix algebras of order two, the Grassmann algebra of plane, and the Kaplansky superalgebra.