Rota–Baxter Operators on Quadratic Algebras
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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.
KeywordsRota–Baxter operator Yang–Baxter equation Quadratic algebra Grassmann algebra Jordan algebra of bilinear form Matrix algebra Kaplansky superalgebra
Mathematics Subject ClassificationPrimary 17A45 Secondary 17C50 Thirdly 16T25
Open access funding provided by Austrian Science Fund (FWF). Pilar Benito acknowledges financial support by Grant MTM2017-83506-C2-1-P (AEI/FEDER, UE). Vsevolod Gubarev is supported by the Austrian Science Foundation FWF, Grant P28079.
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