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Multiple Rota-Baxter algebra and multiple Rota-Baxter modules

Jun He, Xiaosong Peng, Yi Zhang

TL;DR

This work develops a comprehensive framework for multiple Rota-Baxter structures by introducing left, right, and bimodule MRB modules over a multi-operator algebra $(R,P_{\Omega})$ with pair weight $(\lambda_{\Omega},\lambda_{\Omega})$, and by formalizing the theory of free, projective, injective, and flat MRB modules. It constructs free MRB modules via $\Omega$-operated modules and imposes relations to enforce MRB identities, establishing universal properties and conditions for freeness; it further extends to restricted free modules using MC$(M)$ and demonstrates isomorphisms with standard free modules in singleton cases. The paper develops a Hom functor $\mathrm{Hom}_{(R,P_{\Omega})}$, proves the category has enough projectives and injectives, and introduces the ring of multiple Rota-Baxter operators $R_{MRB}\langle Q_{\Omega}\rangle$ to relate MRB-modules to module categories over this ring. A tensor product over MRB algebras is defined with a universal property, and flatness is established by showing that free and projective MRB-modules are flat and that tensoring preserves exact sequences. Overall, the results equip multi-operator Rota-Baxter theory with a robust homological toolkit, enabling derived functors, tensorial constructions, and flatness theory in a multi-operator setting.

Abstract

In this paper, we develop the theory of multiple Rota-Baxter modules over multiple Rota-Baxter algebras. We introduce left, right, and bimodule structures and construct free $Ω$-operated modules with mixable tensor establishing free commutative multiple Rota-Baxter modules. We provide a necessary and sufficient condition for a free module to admit a free multiple Rota-Baxter module structure. Furthermore, we define projective and injective multiple Rota-Baxter modules, showing that their category has enough projective and injective objects to support derived $\mathrm{Hom}$ functors. Finally, we introduce the tensor product of multiple Rota-Baxter algebras and define flat multiple Rota-Baxter modules, proving that both free and projective modules satisfy the flatness property.

Multiple Rota-Baxter algebra and multiple Rota-Baxter modules

TL;DR

This work develops a comprehensive framework for multiple Rota-Baxter structures by introducing left, right, and bimodule MRB modules over a multi-operator algebra with pair weight , and by formalizing the theory of free, projective, injective, and flat MRB modules. It constructs free MRB modules via -operated modules and imposes relations to enforce MRB identities, establishing universal properties and conditions for freeness; it further extends to restricted free modules using MC and demonstrates isomorphisms with standard free modules in singleton cases. The paper develops a Hom functor , proves the category has enough projectives and injectives, and introduces the ring of multiple Rota-Baxter operators to relate MRB-modules to module categories over this ring. A tensor product over MRB algebras is defined with a universal property, and flatness is established by showing that free and projective MRB-modules are flat and that tensoring preserves exact sequences. Overall, the results equip multi-operator Rota-Baxter theory with a robust homological toolkit, enabling derived functors, tensorial constructions, and flatness theory in a multi-operator setting.

Abstract

In this paper, we develop the theory of multiple Rota-Baxter modules over multiple Rota-Baxter algebras. We introduce left, right, and bimodule structures and construct free -operated modules with mixable tensor establishing free commutative multiple Rota-Baxter modules. We provide a necessary and sufficient condition for a free module to admit a free multiple Rota-Baxter module structure. Furthermore, we define projective and injective multiple Rota-Baxter modules, showing that their category has enough projective and injective objects to support derived functors. Finally, we introduce the tensor product of multiple Rota-Baxter algebras and define flat multiple Rota-Baxter modules, proving that both free and projective modules satisfy the flatness property.

Paper Structure

This paper contains 13 sections, 26 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Rota-Baxter algebras and Rota-Baxter modules
  3. Rota-Baxter algebras (modules) going multiple
  4. Free multiple Rota-Baxter modules
  5. Multiple Rota-Baxter modules
  6. Free multiple Rota-Baxter modules
  7. Free modules as free multiple Rota-Baxter modules
  8. Projective and injective multiple Rota-Baxter module
  9. The Hom functor
  10. Projective and injective multiple Rota-Baxter module
  11. Flat multiple Rota-Baxter modules
  12. Tensor product of multiple Rota-Baxter modules
  13. Flat multiple Rota-Baxter module

Key Result

Lemma 2.4

Let $(M,\mathfrak{m}_{\Omega})$ be a left $(R,P_{\Omega})$-module and $(N,\mathfrak{n}_{\Omega})$ a left $(R,P_{\Omega})$ submodule of $(M,\mathfrak{m}_{\Omega})$. For each $\omega\in\Omega$, define Then $(M/N,\overline{\mathfrak{m}}_{\Omega})$ is a left $(R,P_{\Omega})$-module with $\overline{\mathfrak{m}}_{\Omega}:=(\overline{\mathfrak{m}}_{\omega})_{\omega\in\Omega}$. We refer to $(M/N,\overli

Theorems & Definitions (63)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Example 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Definition 2.8
  • ...and 53 more