Leibniz-dendriform bialgebras and relative Rota-Baxter operators

2025/10/19 by Sun, Qinxiu, Guo, Shuangjian
#FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2510.16826

Abstract

In this paper, we introduce the notion of Leibniz-dendriform bialgebras and establish their equivalence with phase spaces and matched pairs of Leibniz algebras. The study of the coboundary case leads naturally to the Leibniz-dendriform Yang-Baxter equation (LD-YBE). We prove that skew-symmetric solutions of the LD-YBE give rise to coboundary Leibniz-dendriform bialgebras. Furthermore, we demonstrate that solutions not necessarily skew-symmetric can also induce such bialgebras. This observation motivates the introduction of quasi-triangular and factorizable Leibniz-dendriform bialgebras. In particular, we show that solutions of the LD-YBE with invariant symmetric parts yield quasi-triangular Leibniz-dendriform bialgebras. Such solutions are also interpreted as relative Rota-Baxter operators with weights. Finally, we establish a one-to-one correspondence between quadratic Rota-Baxter Leibniz-dendriform algebras and factorizable Leibniz-dendriform bialgebras.

Citations

• Extending structures for anti-dendriform algebras and anti-dendriform bialgebras
• Quasi-triangular, factorizable Leibniz bialgebras and relative Rota–Baxter operators
• Quasi-triangular pre-Lie bialgebras, factorizable pre-Lie bialgebras and Rota-Baxter pre-Lie algebras
• Quasi-Frobenius Novikov algebras and pre-Novikov bialgebras
• Quasi-Frobenius Novikov algebras and pre-Novikov bialgebras
• Symplectic structure, product structures and complex structures on Leibniz algebras
• Extending structures for perm algebras and perm bialgebras
• Extending structures for perm algebras and perm bialgebras
• Quasi-triangular and factorizable antisymmetric infinitesimal bialgebras
• Pre-Leibniz algebras
• Factorizable Lie Bialgebras, Quadratic Rota–Baxter Lie Algebras and Rota–Baxter Lie Bialgebras
• On quadri-algebras
• Leibniz homology of Lie algebras as functor homology
• Integrable Systems and Factorization Problems
• On the Associative Analog of Lie Bialgebras
• Universal enveloping algebras of Leibniz algebras and (co)homology
• Quantum R-matrices and factorization problems

Related

• Quasi-triangular, factorizable Leibniz bialgebras and relative Rota–Baxter operators
• Quasi-triangular, factorizable anti-dendriform bialgebras and relative Rota-Baxter operators
• Quasi-triangular, triangular, factorizable anti-Leibniz bialgebras and anti-Leibniz Yang-Baxter equation
• Quasi-triangular and factorizable dendriform D-bialgebras
• Rota-Baxter Lie bialgebras, classical Yang-Baxter equations and special L-dendriform bialgebras