Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.12239 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909844349386752 |
|---|---|
| author | Foissy, Loïc Xie, Yunzhou Zhang, Dawei Zhang, Yi |
| author_facet | Foissy, Loïc Xie, Yunzhou Zhang, Dawei Zhang, Yi |
| contents | The concept of weighted infinitesimal bialgebras provides an algebraic framework for understanding the non-homogeneous associative Yang-Baxter equation. In this paper, we endow the space of decorated planar rooted forests with a two-parameters family of coproducts, making it into a weighted infinitesimal bialgebra. A combinatorial characterization of the coproducts is given via the notion of forest biideals. Furthermore, by constructing a bilinear symmetric form and introducing a new grafting operation on rooted forests, we describe the associated dual products. We also introduce the notion of the pair-weight 1-cocycle condition and investigate the universal properties of decorated planar rooted forests satisfying this condition. This leads to the definition of a weighted $Ω$-cocycle infinitesimal unitary bialgebra. As applications, we identify the initial object in the category of free cocycle infinitesimal unitary bialgebras on undecorated planar rooted forests, corresponding to the well-known noncommutative Connes-Kreimer Hopf algebra. In addition, we establish isomorphisms between different coproduct structures and construct a pre-Lie algebra structure on decorated planar rooted forests. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12239 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cocycle weighted infinitesimal bialgebras and pre-Lie algebras on rooted trees Foissy, Loïc Xie, Yunzhou Zhang, Dawei Zhang, Yi Combinatorics The concept of weighted infinitesimal bialgebras provides an algebraic framework for understanding the non-homogeneous associative Yang-Baxter equation. In this paper, we endow the space of decorated planar rooted forests with a two-parameters family of coproducts, making it into a weighted infinitesimal bialgebra. A combinatorial characterization of the coproducts is given via the notion of forest biideals. Furthermore, by constructing a bilinear symmetric form and introducing a new grafting operation on rooted forests, we describe the associated dual products. We also introduce the notion of the pair-weight 1-cocycle condition and investigate the universal properties of decorated planar rooted forests satisfying this condition. This leads to the definition of a weighted $Ω$-cocycle infinitesimal unitary bialgebra. As applications, we identify the initial object in the category of free cocycle infinitesimal unitary bialgebras on undecorated planar rooted forests, corresponding to the well-known noncommutative Connes-Kreimer Hopf algebra. In addition, we establish isomorphisms between different coproduct structures and construct a pre-Lie algebra structure on decorated planar rooted forests. |
| title | Cocycle weighted infinitesimal bialgebras and pre-Lie algebras on rooted trees |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.12239 |