A functorial approach to Kashiwara-Vergne
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918230792077312 |
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| author | Navarro-Betancourt, Rodrigo |
| author_facet | Navarro-Betancourt, Rodrigo |
| contents | As a consequence of the proof of the Kashiwara-Vergne conjecture of Alekseev and Torossian, the authors obtained an injection $\mathrm{GRT} \hookrightarrow \mathrm{KRV}$. The group $\mathrm{GRT}$ can be regarded as the group of automorphisms of the operad of parenthesized chord diagrams, while $\mathrm{KRV}$ can be recovered from the automorphism group of the Goldman-Turaev Lie bialgebra of a thrice-punctured sphere. This suggests the existence of a natural way to derive Lie bialgebras from operads, and we verify this is the case. That is, we reproduce the Alekseev-Torossian injection by functorially constructing bracket and cobracket operations out of operad modules. This framework is enough to establish a relationship between Gonzalez' higher genus $\mathrm{GRT}_g$ groups, and the higher genus $\mathrm{KRV}_g$ groups of Alekseev, Kawazumi, Kuno, and Naef. Our construction is informed by Massuyeau and Turaev's work on Fox pairings and quasi-derivations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04091 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A functorial approach to Kashiwara-Vergne Navarro-Betancourt, Rodrigo Quantum Algebra Geometric Topology As a consequence of the proof of the Kashiwara-Vergne conjecture of Alekseev and Torossian, the authors obtained an injection $\mathrm{GRT} \hookrightarrow \mathrm{KRV}$. The group $\mathrm{GRT}$ can be regarded as the group of automorphisms of the operad of parenthesized chord diagrams, while $\mathrm{KRV}$ can be recovered from the automorphism group of the Goldman-Turaev Lie bialgebra of a thrice-punctured sphere. This suggests the existence of a natural way to derive Lie bialgebras from operads, and we verify this is the case. That is, we reproduce the Alekseev-Torossian injection by functorially constructing bracket and cobracket operations out of operad modules. This framework is enough to establish a relationship between Gonzalez' higher genus $\mathrm{GRT}_g$ groups, and the higher genus $\mathrm{KRV}_g$ groups of Alekseev, Kawazumi, Kuno, and Naef. Our construction is informed by Massuyeau and Turaev's work on Fox pairings and quasi-derivations. |
| title | A functorial approach to Kashiwara-Vergne |
| topic | Quantum Algebra Geometric Topology |
| url | https://arxiv.org/abs/2512.04091 |