## Abstract

The Rota-Baxter algebra and the shuffle product are both algebraic structures arising from integral operators and integral equations. Free commutative Rota-Baxter algebras provide an algebraic framework for integral equations with the simple Riemann integral operator. The Zinbiel algebras form a category in which the shuffle product algebra is the free object. Motivated by algebraic structures underlying integral equations involving multiple integral operators and kernels, we study commutative matching Rota-Baxter algebras and construct the free objects making use of the shuffle product with multiple decorations. We also construct free commutative matching Rota-Baxter algebras in a relative context, to emulate the action of the integral operators on the coefficient functions in an integral equation. We finally show that free commutative matching Rota-Baxter algebras give the free matching Zinbiel algebra, generalizing the characterization of the shuffle product algebra as the free Zinbiel algebra obtained by Loday.

Original language | English (US) |
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Pages (from-to) | 402-432 |

Number of pages | 31 |

Journal | Journal of Algebra |

Volume | 586 |

DOIs | |

State | Published - Nov 15 2021 |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

## Keywords

- Matching Rota-Baxter algebra
- Matching Zinbiel algebra
- Matching dendriform algebra
- Shuffle product with decoration