On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911708837052416 |
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| author | Li, Yunnan |
| author_facet | Li, Yunnan |
| contents | Recently the notion of post-Hopf algebra was introduced, with the universal enveloping algebra of a post-Lie algebra as the fundamental example. A novel property is that any cocommutative post-Hopf algebra gives rise to a sub-adjacent Hopf algebra with a generalized Grossman-Larson product. By twisting the post-Hopf product, we provide a combinatorial antipode formula for the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra. Relating to such a sub-adjacent Hopf algebra, we also obtain a closed inverse formula for the Oudom-Guin isomorphism in the context of post-Lie algebras. Especially as a byproduct, we derive a cancellation-free antipode formula for the Grossman-Larson Hopf algebra of ordered trees through a concrete tree-grafting expression. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01345 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra Li, Yunnan Rings and Algebras Mathematical Physics Combinatorics Quantum Algebra Recently the notion of post-Hopf algebra was introduced, with the universal enveloping algebra of a post-Lie algebra as the fundamental example. A novel property is that any cocommutative post-Hopf algebra gives rise to a sub-adjacent Hopf algebra with a generalized Grossman-Larson product. By twisting the post-Hopf product, we provide a combinatorial antipode formula for the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra. Relating to such a sub-adjacent Hopf algebra, we also obtain a closed inverse formula for the Oudom-Guin isomorphism in the context of post-Lie algebras. Especially as a byproduct, we derive a cancellation-free antipode formula for the Grossman-Larson Hopf algebra of ordered trees through a concrete tree-grafting expression. |
| title | On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra |
| topic | Rings and Algebras Mathematical Physics Combinatorics Quantum Algebra |
| url | https://arxiv.org/abs/2408.01345 |