Hypertrees and embedding of the $\mathrm{FMan}$ operad
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910311979679744 |
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| author | Laubie, Paul |
| author_facet | Laubie, Paul |
| contents | The operad $\mathrm{FMan}$ encodes the algebraic structure on vector fields of Frobenius manifolds, in the same way as the operad $\mathrm{Lie}$ encodes the algebraic structure on vector fields of a smooth manifold. It is well known that the operad $\mathrm{Lie}$ admits an embedding in the operad $\mathrm{PreLie}$ encoding pre-Lie algebras. We prove a conjecture of Dotsenko stating that the operad $\mathrm{FMan}$ admits an embedding in the operad $\mathrm{ComPreLie}$. The operad $\mathrm{ComPreLie}$ is the operad encoding pre-Lie algebras with an additional commutative product such that right pre-Lie multiplications act as derivations. To prove this result, we first remark a link between the Greg trees and the so-called operadic twisting of $\mathrm{PreLie}$. We then give a combinatorial description of the operad $\mathrm{ComPreLie}$ \emph{à la} Chapoton-Livernet with forests of rooted hypertrees. We generalize this construction to forests of rooted Greg hypertrees, and then use operadic twisting techniques to prove the conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17439 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hypertrees and embedding of the $\mathrm{FMan}$ operad Laubie, Paul Quantum Algebra Combinatorics K-Theory and Homology The operad $\mathrm{FMan}$ encodes the algebraic structure on vector fields of Frobenius manifolds, in the same way as the operad $\mathrm{Lie}$ encodes the algebraic structure on vector fields of a smooth manifold. It is well known that the operad $\mathrm{Lie}$ admits an embedding in the operad $\mathrm{PreLie}$ encoding pre-Lie algebras. We prove a conjecture of Dotsenko stating that the operad $\mathrm{FMan}$ admits an embedding in the operad $\mathrm{ComPreLie}$. The operad $\mathrm{ComPreLie}$ is the operad encoding pre-Lie algebras with an additional commutative product such that right pre-Lie multiplications act as derivations. To prove this result, we first remark a link between the Greg trees and the so-called operadic twisting of $\mathrm{PreLie}$. We then give a combinatorial description of the operad $\mathrm{ComPreLie}$ \emph{à la} Chapoton-Livernet with forests of rooted hypertrees. We generalize this construction to forests of rooted Greg hypertrees, and then use operadic twisting techniques to prove the conjecture. |
| title | Hypertrees and embedding of the $\mathrm{FMan}$ operad |
| topic | Quantum Algebra Combinatorics K-Theory and Homology |
| url | https://arxiv.org/abs/2401.17439 |