Homotopy theory of post-Lie algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913810577620992 |
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| author | Lazarev, Andrey Sheng, Yunhe Tang, Rong |
| author_facet | Lazarev, Andrey Sheng, Yunhe Tang, Rong |
| contents | In this paper, we study the homotopy theory of post-Lie algebras. Guided by Koszul duality theory, we consider the graded Lie algebra of coderivations of the cofree conilpotent graded cocommutative cotrialgebra generated by $V$. We show that in the case of $V$ being a shift of an ungraded vector space $W$, Maurer-Cartan elements of this graded Lie algebra are exactly post-Lie algebra structures on $W$. The cohomology of a post-Lie algebra is then defined using Maurer-Cartan twisting. The second cohomology group of a post-Lie algebra has a familiar interpretation as equivalence classes of infinitesimal deformations. Next we define a post-Lie$_\infty$ algebra structure on a graded vector space to be a Maurer-Cartan element of the aforementioned graded Lie algebra. Post-Lie$_\infty$ algebras admit a useful characterization in terms of $L_\infty$-actions (or open-closed homotopy Lie algebras). Finally, we introduce the notion of homotopy Rota-Baxter operators on open-closed homotopy Lie algebras and show that certain homotopy Rota-Baxter operators induce post-Lie$_\infty$ algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_19998 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homotopy theory of post-Lie algebras Lazarev, Andrey Sheng, Yunhe Tang, Rong Rings and Algebras In this paper, we study the homotopy theory of post-Lie algebras. Guided by Koszul duality theory, we consider the graded Lie algebra of coderivations of the cofree conilpotent graded cocommutative cotrialgebra generated by $V$. We show that in the case of $V$ being a shift of an ungraded vector space $W$, Maurer-Cartan elements of this graded Lie algebra are exactly post-Lie algebra structures on $W$. The cohomology of a post-Lie algebra is then defined using Maurer-Cartan twisting. The second cohomology group of a post-Lie algebra has a familiar interpretation as equivalence classes of infinitesimal deformations. Next we define a post-Lie$_\infty$ algebra structure on a graded vector space to be a Maurer-Cartan element of the aforementioned graded Lie algebra. Post-Lie$_\infty$ algebras admit a useful characterization in terms of $L_\infty$-actions (or open-closed homotopy Lie algebras). Finally, we introduce the notion of homotopy Rota-Baxter operators on open-closed homotopy Lie algebras and show that certain homotopy Rota-Baxter operators induce post-Lie$_\infty$ algebras. |
| title | Homotopy theory of post-Lie algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2504.19998 |