Homotopy theory of post-Lie algebras

Andrey Lazarev; Yunhe Sheng; Rong Tang

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arXiv · 2504.19998

Homotopy theory of post-Lie algebras

Andrey Lazarev·Yunhe Sheng·Rong Tang

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Abstract

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.

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Andrey Lazarev, Yunhe Sheng, Rong Tang. 2025-04-28. Homotopy theory of post-Lie algebras. https://arxiv.org/abs/2504.19998

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