Genus Zero Kashiwara-Vergne Solutions from Braids
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915916896272384 |
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| author | Dancso, Zsuzsanna Halacheva, Iva Laplante-Anfossi, Guillaume Robertson, Marcy Singh, Chandan |
| author_facet | Dancso, Zsuzsanna Halacheva, Iva Laplante-Anfossi, Guillaume Robertson, Marcy Singh, Chandan |
| contents | Using the language of moperads -- monoids in the category of right modules over an operad -- we reinterpret the Alekseev--Enriquez--Torossian construction of Kashiwara--Vergne (KV) solutions from associators. We show that any equivalence between the moperad of parenthesized braids with a frozen strand and the moperad of chord diagrams gives rise to a family of genus zero KV solutions operadically generated by a single classical KV solution. We show that the Grothendieck--Teichmüller module groups act on the latter, intertwining the actions of the KV symmetry groups. In the other direction, we show that any symmetric KV solution gives rise to a module map from parenthesized braids with a frozen strand to tangential automorphisms of free Lie algebras. This map factors through the moperad of chord diagrams if and only if the associated KV associator is a Drinfeld associator. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_16243 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Genus Zero Kashiwara-Vergne Solutions from Braids Dancso, Zsuzsanna Halacheva, Iva Laplante-Anfossi, Guillaume Robertson, Marcy Singh, Chandan Algebraic Topology Category Theory Quantum Algebra 18M60, 17B, 55 Using the language of moperads -- monoids in the category of right modules over an operad -- we reinterpret the Alekseev--Enriquez--Torossian construction of Kashiwara--Vergne (KV) solutions from associators. We show that any equivalence between the moperad of parenthesized braids with a frozen strand and the moperad of chord diagrams gives rise to a family of genus zero KV solutions operadically generated by a single classical KV solution. We show that the Grothendieck--Teichmüller module groups act on the latter, intertwining the actions of the KV symmetry groups. In the other direction, we show that any symmetric KV solution gives rise to a module map from parenthesized braids with a frozen strand to tangential automorphisms of free Lie algebras. This map factors through the moperad of chord diagrams if and only if the associated KV associator is a Drinfeld associator. |
| title | Genus Zero Kashiwara-Vergne Solutions from Braids |
| topic | Algebraic Topology Category Theory Quantum Algebra 18M60, 17B, 55 |
| url | https://arxiv.org/abs/2507.16243 |