Pre-anti-flexible bialgebras

Abstract

In this paper, we derive pre-anti-flexible algebras structures in term of zero weight’s Rota-Baxter operators, built underlying left-symmetric algebras, view pre-anti-flexible algebras as a splitting of anti-flexible algebras, introduce pre-anti-flexible bialgebras and establish its equivalences among matched pair of anti-flexible algebras and matched pair of pre-anti-flexible algebras. Special class of pre-anti-flexible bialgebras leads to the establishment of the pre-anti-flexible Yang-Baxter equation which is the same with $\mathcal{D}$-equation. Symmetric solution of pre-anti-flexible Yang-Baxter equation gives a pre-anti-flexible bialgebra. Finally, we recall and link $\mathcal{O}$-operators of anti-flexible algebras to bimodules of pre-anti-flexible algebras and built symmetric solutions of anti-flexible Yang-Baxter equation.

Keywords:

• Dendriform algebra
• (pre-)anti-flexible algebra
• (pre-)anti-flexible bialgebra
• Rota-Baxter operator
• Yang-Baxter equation

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17A20
• 17D25
• 16T10
• 17B38
• 16T25

Acknowledgments

The author thanks Professor C. Bai for helpful discussions and his encouragement, and Nankai ZhiDe Foundation.