On post-Lie structures for free Lie algebras
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910920626667520 |
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| author | Burmester, Annika Kühn, Ulf |
| author_facet | Burmester, Annika Kühn, Ulf |
| contents | We study post-Lie structures on free Lie algebras, the Grossman-Larson product on their enveloping algebras, and provide an abstract formula for its dual coproduct. This might be of interest for the general theory of post-Hopf algebras. Using a magmatic approach, we explore post-Lie algebras connected to multiple zeta values and their $q$-analogues. For multiple zeta values, this framework yields an algebraic interpretation of the Goncharov coproduct. Assuming that the Bernoulli numbers satisfy the so called threshold shuffle identities, we present a post-Lie structure, whose induced Lie bracket we expect to restrict to the dual of indecomposables of multiple $q$-zeta values. Our post-Lie algebras align with Ecalle's theory of bimoulds: we explicitly identify the ari bracket with a post-Lie structure on a free Lie algebra, and conjecture a correspondence for the uri bracket. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19661 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On post-Lie structures for free Lie algebras Burmester, Annika Kühn, Ulf Number Theory Mathematical Physics Combinatorics Rings and Algebras We study post-Lie structures on free Lie algebras, the Grossman-Larson product on their enveloping algebras, and provide an abstract formula for its dual coproduct. This might be of interest for the general theory of post-Hopf algebras. Using a magmatic approach, we explore post-Lie algebras connected to multiple zeta values and their $q$-analogues. For multiple zeta values, this framework yields an algebraic interpretation of the Goncharov coproduct. Assuming that the Bernoulli numbers satisfy the so called threshold shuffle identities, we present a post-Lie structure, whose induced Lie bracket we expect to restrict to the dual of indecomposables of multiple $q$-zeta values. Our post-Lie algebras align with Ecalle's theory of bimoulds: we explicitly identify the ari bracket with a post-Lie structure on a free Lie algebra, and conjecture a correspondence for the uri bracket. |
| title | On post-Lie structures for free Lie algebras |
| topic | Number Theory Mathematical Physics Combinatorics Rings and Algebras |
| url | https://arxiv.org/abs/2504.19661 |