Splitting of operations, Manin products and Rota-Baxter operators

TL;DR

This paper introduces a new operadic framework for splitting algebraic operations, linking it to Manin products and Rota-Baxter operators, and applies it to matrix algebra structures with various examples.

Contribution

It provides a novel operadic approach to splitting operations, establishing equivalences with Manin products and Rota-Baxter operators, and explores symmetry and matrix algebra applications.

Findings

Operadic definition of operation splitting established

Equivalence with Manin products demonstrated

Applications to matrix algebra and Jordan algebras shown

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 and it is shown to be closely related to Rota-Baxter operators. Hence, it gives a new effective way to compute Manin black products. The present construction is shown to have symmetry properties. 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 the case of Jordan algebras.