2025/04/28 by Andrey Lazarev, Yunhe Sheng, Lazarev, Andrey +3 · 1 citation
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2504.19998
openalex publication_date 2025/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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_∞ algebra structure on a graded vector space to be a Maurer-Cartan element of the aforementioned graded Lie algebra. Post-Lie_∞ algebras admit a useful characterization in terms of L_∞-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_∞ algebras.