Mathematical Physics
[Submitted on 23 Mar 2022]
Title:Post-Hopf algebras, relative Rota-Baxter operators and solutions of the Yang-Baxter equation
View PDFAbstract:In this paper, first we introduce the notion of a post-Hopf algebra, which gives rise to a post-Lie algebra on the space of primitive elements and there is naturally a post-Hopf algebra structure on the universal enveloping algebra of a post-Lie algebra. A novel property is that a cocommutative post-Hopf algebra gives rise to a generalized Grossman-Larsson product, which leads to a subadjacent Hopf algebra and can be used to construct solutions of the Yang-Baxter equation. Then we introduce the notion of relative Rota-Baxter operators on Hopf algebras. A cocommutative post-Hopf algebra gives rise to a relative Rota-Baxter operator on its subadjacent Hopf algebra. Conversely, a relative Rota-Baxter operator also induces a post-Hopf algebra. Then we show that relative Rota-Baxter operators give rise to matched pairs of Hopf algebras. Consequently, post-Hopf algebras and relative Rota-Baxter operators give solutions of the Yang-Baxter equation in certain cocommutative Hopf algebras. Finally we characterize relative Rota-Baxter operators on Hopf algebras using relative Rota-Baxter operators on the Lie algebra of primitive elements, graphs and module bialgebra structures.
Current browse context:
math-ph
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.