Rota-Baxter Operators on Quadratic Algebras

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.

23 p

References in corpus (3)

• Loday-type Algebras and the Rota-Baxter Relation
• From Rota-Baxter Algebras to Pre-Lie Algebras
• Quasi-idempotent Rota-Baxter operators arising from quasi-idempotent elements

Cited by in corpus (12)

• Rota-Baxter operators on unital algebras
• Rota-Baxter operators on a sum of fields
• Spectrum of Rota-Baxter operators
• Rota-type operators on null-filiform associative algebras
• Rota-Baxter operators of weight zero on upper-triangular matrices of order three
• Rota-Baxter operators of nonzero weight on the split Cayley-Dickson algebra
• Rota-Baxter operators on the simple Jordan algebra of matrices of order two
• Rota-Baxter operators of weight zero on the matrix algebra of order three without unit in kernel
• Monomial Rota-Baxter operators on free commutative non-unital algebra
• Rota-Baxter operators on $\mathrm{Cur}({\mathrm{sl}}_2(\mathbb{C}))$
• Double Lie algebras of a nonzero weight
• Rota-Baxter operators of nonzero weight on the matrix algebra of order three