Splitting of Operations, Manin Products, and Rota–Baxter Operators

Abstract

This paper provides a general operadic definition for the notion of splitting the operations of algebraic structures. This construction is proved to be equivalent to some Manin products of operads in the case of quadratic operads and it is shown to be closely related to Rota–Baxter operators. Hence, it gives a new effective way to compute Manin black products. Finally, this allows us to describe the algebraic structure of square matrices with coefficients in algebras of certain types. Many examples illustrate this text, including an example of nonquadratic algebras with Jordan algebras.