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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2406.12559 |
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| _version_ | 1866913396760248320 |
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| author | Giraudo, Samuele |
| author_facet | Giraudo, Samuele |
| contents | The natural Hopf algebra $\mathbf{N} \cdot \mathcal{O}$ of an operad $\mathcal{O}$ is a Hopf algebra whose bases are indexed by some words on $\mathcal{O}$. We construct polynomial realizations of $\mathbf{N} \cdot \mathcal{O}$ by using alphabets of noncommutative variables endowed with unary and binary relations. By using particular alphabets, we establish links between $\mathbf{N} \cdot \mathcal{O}$ and some other Hopf algebras including the Hopf algebra of word quasi-symmetric functions of Hivert, the decorated versions of the noncommutative Connes-Kreimer Hopf algebra of Foissy, the noncommutative Faà di Bruno Hopf algebra and its deformations, the noncommutative multi-symmetric functions Hopf algebras of Novelli and Thibon, and the double tensor Hopf algebra of Ebrahimi-Fard and Patras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12559 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polynomial realizations of Hopf algebras built from nonsymmetric operads Giraudo, Samuele Combinatorics Rings and Algebras 16T30, 18M80, 05E05 The natural Hopf algebra $\mathbf{N} \cdot \mathcal{O}$ of an operad $\mathcal{O}$ is a Hopf algebra whose bases are indexed by some words on $\mathcal{O}$. We construct polynomial realizations of $\mathbf{N} \cdot \mathcal{O}$ by using alphabets of noncommutative variables endowed with unary and binary relations. By using particular alphabets, we establish links between $\mathbf{N} \cdot \mathcal{O}$ and some other Hopf algebras including the Hopf algebra of word quasi-symmetric functions of Hivert, the decorated versions of the noncommutative Connes-Kreimer Hopf algebra of Foissy, the noncommutative Faà di Bruno Hopf algebra and its deformations, the noncommutative multi-symmetric functions Hopf algebras of Novelli and Thibon, and the double tensor Hopf algebra of Ebrahimi-Fard and Patras. |
| title | Polynomial realizations of Hopf algebras built from nonsymmetric operads |
| topic | Combinatorics Rings and Algebras 16T30, 18M80, 05E05 |
| url | https://arxiv.org/abs/2406.12559 |