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Rota-Baxter operators on quadratic algebras. (English) Zbl 1431.17003
Summary: 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.

##### MSC:
 17A45 Quadratic algebras (but not quadratic Jordan algebras) 17C50 Jordan structures associated with other structures 16T25 Yang-Baxter equations
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