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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.13317 |
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| _version_ | 1866929274525581312 |
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| author | Foissy, Loïc Patras, Frédéric |
| author_facet | Foissy, Loïc Patras, Frédéric |
| contents | We here construct an explicit isomorphism between any commutative Hopf algebra which underlying coalgebra is the tensor coalgebra of a space $V$ and the shuffle algebra based on the same space. This isomorphism uses the commutative $B_\infty$ structure that governs the product and the eulerian idempotent, as well as the canonical projection on the space $V$. This generalizes Homan's isomorphism between commutative quasi-shuffle and shuffle algebras, which correspond to the case when the $B_\infty$ structure is given by an associative and commutative product. We develop several examples in details, including the Hopf algebra of finite topologies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13317 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Commutative B\_infty -algebras are shuffle algebras Foissy, Loïc Patras, Frédéric Combinatorics We here construct an explicit isomorphism between any commutative Hopf algebra which underlying coalgebra is the tensor coalgebra of a space $V$ and the shuffle algebra based on the same space. This isomorphism uses the commutative $B_\infty$ structure that governs the product and the eulerian idempotent, as well as the canonical projection on the space $V$. This generalizes Homan's isomorphism between commutative quasi-shuffle and shuffle algebras, which correspond to the case when the $B_\infty$ structure is given by an associative and commutative product. We develop several examples in details, including the Hopf algebra of finite topologies. |
| title | Commutative B\_infty -algebras are shuffle algebras |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2401.13317 |