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Symmetric Rota-Baxter systems and applications

Chan Zhao, Haiying Li, Tianshui Ma

Abstract

Rota-Baxter operators and bialgebras are closely connected in several applications, such as the Connes-Kreimer renormalization framework and the operator approach to the classical Yang-Baxter equation. The concept of a Rota-Baxter system was introduced in 2016 as a generalization of a Rota-Baxter operator. In this work, we introduce a bialgebra structure compatible with a symmetric Rota-Baxter system, called a symmetric Rota-Baxter antisymmetric infinitesimal (ASI) bisystem. This bialgebra is characterized by generalizations of matched pairs of algebras and double constructions of Frobenius algebras to the setting of symmetric Rota-Baxter systems. Investigating the coboundary case leads to an enriched version of the associative Yang-Baxter equation (aYBe) adapted to symmetric Rota-Baxter systems. Antisymmetric solutions of this equation are used to construct symmetric Rota-Baxter ASI bisystems. We also introduce the notion of an $\mathcal{O}$-operator on a symmetric Rota-Baxter system, which produces solutions of the admissible aYBe in such systems and thereby gives rise to symmetric Rota-Baxter ASI bisystems. A symmetric Rota-Baxter bisystem generalizes several known structures, including Rota-Baxter Lie bisystems, Rota-Baxter ASI bialgebras, Rota-Baxter Lie bialgebras, averaging ASI bialgebras, averaging Lie bialgebras, and special apre-perm bialgebras.

Symmetric Rota-Baxter systems and applications

Abstract

Rota-Baxter operators and bialgebras are closely connected in several applications, such as the Connes-Kreimer renormalization framework and the operator approach to the classical Yang-Baxter equation. The concept of a Rota-Baxter system was introduced in 2016 as a generalization of a Rota-Baxter operator. In this work, we introduce a bialgebra structure compatible with a symmetric Rota-Baxter system, called a symmetric Rota-Baxter antisymmetric infinitesimal (ASI) bisystem. This bialgebra is characterized by generalizations of matched pairs of algebras and double constructions of Frobenius algebras to the setting of symmetric Rota-Baxter systems. Investigating the coboundary case leads to an enriched version of the associative Yang-Baxter equation (aYBe) adapted to symmetric Rota-Baxter systems. Antisymmetric solutions of this equation are used to construct symmetric Rota-Baxter ASI bisystems. We also introduce the notion of an -operator on a symmetric Rota-Baxter system, which produces solutions of the admissible aYBe in such systems and thereby gives rise to symmetric Rota-Baxter ASI bisystems. A symmetric Rota-Baxter bisystem generalizes several known structures, including Rota-Baxter Lie bisystems, Rota-Baxter ASI bialgebras, Rota-Baxter Lie bialgebras, averaging ASI bialgebras, averaging Lie bialgebras, and special apre-perm bialgebras.
Paper Structure (23 sections, 48 theorems, 88 equations)

This paper contains 23 sections, 48 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Rota-Baxter systems
  3. Bialgebras
  4. Main results and layout of the paper
  5. Symmetric Rota-Baxter systems and their representations
  6. Symmetric Rota-Baxter systems
  7. Dual representation
  8. Symmetric Rota-Baxter ASI bisystem
  9. Matched pair of symmetric Rota-Baxter systems
  10. Double construction of symmetric Rota-Baxter Frobenius system
  11. Symmetric Rota-Baxter ASI bisystems
  12. Admissible associative Yang-Baxter equations and $\mathcal{O}$-operators
  13. Admissible associative Yang-Baxter equations
  14. $\mathcal{O}$-operator
  15. Quasitriangular case
  16. ...and 8 more sections

Key Result

Lemma 2.5

Let $A$ be an algebra and $\lambda\in K$. Then $R$ is a Rota-Baxter operator of weight $\lambda$ on $A$ if and only if $(A, R, R+\lambda {\rm id}_{A})$ or $(A, R+\lambda {\rm id}_{A}, R)$ is a symmetric Rota-Baxter system.

Theorems & Definitions (135)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Remark 2.7
  • ...and 125 more