Moerdijk Hopf algebras of decorated rooted forests: an operated algebra approach
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914005908455424 |
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| author | Foissy, Loic Peng, Xiao-Song Xie, Yunzhou Zhang, Yi |
| author_facet | Foissy, Loic Peng, Xiao-Song Xie, Yunzhou Zhang, Yi |
| contents | In this paper, we first endow the space of decorated planar rooted forests with a coproduct that equips it with the structure of a bialgebra and further a Moerdijk Hopf algebra. We also present a combinatorial description of this coproduct, and further give an explicit formulation of its dual coproducts through the newly defined notion of forest-representable matrices. By viewing the Moerdijk Hopf algebra within the framework of operated algebras, we introduce the notion of a multiple cocycle Hopf algebra, incorporating a symmetric Hochschild 1-cocycle condition. We then show that the antipode of this Hopf algebra is a Rota-Baxter operator on Moerdijk Hopf algebras. Furthermore, we investigate the universal properties of cocycle Hopf algebras. As an application, we construct the initial object in the category of free cocycle Hopf algebras on undecorated planar rooted forests, which coincides with the well-known Moerdijk Hopf algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18658 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Moerdijk Hopf algebras of decorated rooted forests: an operated algebra approach Foissy, Loic Peng, Xiao-Song Xie, Yunzhou Zhang, Yi Rings and Algebras 16W99, 05C05, 16S10, 16T10, 16T30, 17B60 In this paper, we first endow the space of decorated planar rooted forests with a coproduct that equips it with the structure of a bialgebra and further a Moerdijk Hopf algebra. We also present a combinatorial description of this coproduct, and further give an explicit formulation of its dual coproducts through the newly defined notion of forest-representable matrices. By viewing the Moerdijk Hopf algebra within the framework of operated algebras, we introduce the notion of a multiple cocycle Hopf algebra, incorporating a symmetric Hochschild 1-cocycle condition. We then show that the antipode of this Hopf algebra is a Rota-Baxter operator on Moerdijk Hopf algebras. Furthermore, we investigate the universal properties of cocycle Hopf algebras. As an application, we construct the initial object in the category of free cocycle Hopf algebras on undecorated planar rooted forests, which coincides with the well-known Moerdijk Hopf algebra. |
| title | Moerdijk Hopf algebras of decorated rooted forests: an operated algebra approach |
| topic | Rings and Algebras 16W99, 05C05, 16S10, 16T10, 16T30, 17B60 |
| url | https://arxiv.org/abs/2508.18658 |