Some constructions of antisymmetric infinitesimal bialgebras and their applications to Lie bialgebras
Abstract
In this paper, we mainly provide some methods for constructing antisymmetric infinitesimal (ASI) bialgebras and Lie bialgebras using Zinbiel bialgebras and diassociative bialgebras. We first show that there is an ASI bialgebra structure on the tensor product of a diassociative bialgebra and a quadratic Zinbiel algebra, while there is an infinite-dimensional ASI bialgebra structure on the tensor product of a Zinbiel bialgebra and a quadratic bz-graded diassociative algebra. For a special quadratic bz-graded Leibniz algebra, the property that its tensor product with a Zinbiel bialgebra forms an ASI bialgebra characterizes the Zinbiel bialgebra. By examining the relationship between solutions of the (classical) Yang-Baxter equation in a Zinbiel algebra and the induced associative algebra, we prove that the induced ASI bialgebra is quasi-triangular (resp. triangular, factorizable) whenever the original Zinbiel bialgebra is quasi-triangular (resp. triangular, factorizable). These conclusions enable us to provide a method for constructing Lie bialgebras from Zinbiel bialgebras and two approaches for constructing Lie bialgebras from diassociative bialgebras. We also provide specific descriptions of the connections between the solutions of the Yang-Baxter equations and the connections between the O-operators corresponding to these constructions.
Related
- Factorizable Lie conformal bialgebras, quadratic Rota-Baxter Lie conformal algebras and some induced structures
- Some constructions of restricted Kakeya sets
- Lyapunov Constructions for System Interconnections Arising from Adaptation in Some Optimization Methods
- Differential perm bialgebras, differential perm Yang-Baxter equation, and their applications in related bialgebra structures
- Some Constructions in the Category of Topological Partial Actions
BibTeX
@misc{hou2026some,
title = {{Some constructions of antisymmetric infinitesimal bialgebras and their applications to Lie bialgebras}},
author = {Bo Hou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27947},
eprint = {2609.27947},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27947},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Some constructions of antisymmetric infinitesimal bialgebras and their applications to Lie bialgebras
AU - Bo Hou
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27947
UR - https://arxiv.org/abs/2609.27947
AB - In this paper, we mainly provide some methods for constructing antisymmetric infinitesimal (ASI) bialgebras and Lie bialgebras using Zinbiel bialgebras and diassociative bialgebras. We first show that there is an ASI bialgebra structure on the tensor product of a diassociative bialgebra and a quadratic Zinbiel algebra, while there is an infinite-dimensional ASI bialgebra structure on the tensor product of a Zinbiel bialgebra and a quadratic bz-graded diassociative algebra. For a special quadratic bz-graded Leibniz algebra, the property that its tensor product with a Zinbiel bialgebra forms an ASI bialgebra characterizes the Zinbiel bialgebra. By examining the relationship between solutions of the (classical) Yang-Baxter equation in a Zinbiel algebra and the induced associative algebra, we prove that the induced ASI bialgebra is quasi-triangular (resp. triangular, factorizable) whenever the original Zinbiel bialgebra is quasi-triangular (resp. triangular, factorizable). These conclusions enable us to provide a method for constructing Lie bialgebras from Zinbiel bialgebras and two approaches for constructing Lie bialgebras from diassociative bialgebras. We also provide specific descriptions of the connections between the solutions of the Yang-Baxter equations and the connections between the O-operators corresponding to these constructions.
ER -