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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2510.18432 |
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| _version_ | 1866912662715105280 |
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| author | Foissy, Loïc |
| author_facet | Foissy, Loïc |
| contents | Noncommutative multi-indices are noncommutative monomials in a $\mathbb{N}$-indexed family of indeterminates. We define on them a $\mathbb{Z}$-graded operadic structure, with the help of a shifting derivation. Multi-indices of degree 0 are called populated: they form a suboperad, isomorphic to the operad of Novikov algebras. This operadic structure, and the relation between pre-Lie and Novikov algebras, induces two bialgebraic structure in cointeraction on commutative multi-indices. We show how to combinatorially embed this double bialgebra into the Connes-Kreimer Hopf algebra of rooted trees, with its two coproducts based, firstly on cuts, secondly, on contraction of edges, and how this embedding can be characterized by a Dyson-Schwinger equation. We also study the unique polynomial invariant compatible with the two bialgebraic structures on multi-indices and use to describe the antipode for the first coproduct. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18432 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Operads and bialgebras of multi-indices, and Novikov algebras Foissy, Loïc Combinatorics Noncommutative multi-indices are noncommutative monomials in a $\mathbb{N}$-indexed family of indeterminates. We define on them a $\mathbb{Z}$-graded operadic structure, with the help of a shifting derivation. Multi-indices of degree 0 are called populated: they form a suboperad, isomorphic to the operad of Novikov algebras. This operadic structure, and the relation between pre-Lie and Novikov algebras, induces two bialgebraic structure in cointeraction on commutative multi-indices. We show how to combinatorially embed this double bialgebra into the Connes-Kreimer Hopf algebra of rooted trees, with its two coproducts based, firstly on cuts, secondly, on contraction of edges, and how this embedding can be characterized by a Dyson-Schwinger equation. We also study the unique polynomial invariant compatible with the two bialgebraic structures on multi-indices and use to describe the antipode for the first coproduct. |
| title | Operads and bialgebras of multi-indices, and Novikov algebras |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.18432 |