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Homotopy theory of post-Lie algebras

Andrey Lazarev, Yunhe Sheng, Rong Tang

TL;DR

The paper develops a comprehensive homotopy theory for $PostLie$ algebras via Koszul duality, identifying the controlling graded Lie algebra of coderivations on the cofree conilpotent cotrialgebra $\mathsf{ComTrias}^c(V)$ and showing that Maurer–Cartan elements correspond to $Post-Lie$ structures on a shifted space. It defines a deformation cohomology through MC twisting, proves that the second cohomology governs infinitesimal deformations, and introduces $PostLie_\infty$ algebras as MC elements that relate to open–closed homotopy Lie algebras and higher geometric structures. A homotopy Rota–Baxter theory on open–closed HTL algebras is developed, with certain operators inducing $PostLie_\infty$ structures, thereby connecting operator theory with higher homotopy post–Lie concepts. The framework yields a robust deformation theory, extensive examples from higher geometric settings, and a natural pathway to constructing homotopy post–Lie structures from $L_\infty$-actions and Rota–Baxter-type data, with potential implications for deformation theory and related geometry.

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.

Homotopy theory of post-Lie algebras

TL;DR

The paper develops a comprehensive homotopy theory for algebras via Koszul duality, identifying the controlling graded Lie algebra of coderivations on the cofree conilpotent cotrialgebra and showing that Maurer–Cartan elements correspond to structures on a shifted space. It defines a deformation cohomology through MC twisting, proves that the second cohomology governs infinitesimal deformations, and introduces algebras as MC elements that relate to open–closed homotopy Lie algebras and higher geometric structures. A homotopy Rota–Baxter theory on open–closed HTL algebras is developed, with certain operators inducing structures, thereby connecting operator theory with higher homotopy post–Lie concepts. The framework yields a robust deformation theory, extensive examples from higher geometric settings, and a natural pathway to constructing homotopy post–Lie structures from -actions and Rota–Baxter-type data, with potential implications for deformation theory and related geometry.

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 . We show that in the case of being a shift of an ungraded vector space , Maurer-Cartan elements of this graded Lie algebra are exactly post-Lie algebra structures on . 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 -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.
Paper Structure (14 sections, 24 theorems, 121 equations)

This paper contains 14 sections, 24 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Homotopy theories
  3. Post-Lie algebras
  4. Rota-Baxter operators
  5. Main results and outline of the paper
  6. Notation and conventions
  7. The controlling algebra of post-Lie algebras
  8. Cohomologies of post-Lie algebras
  9. Post-Lie$_\infty$ algebras
  10. Post-Lie$_\infty$ algebras and open-closed homotopy Lie algebras
  11. Post-Lie$_\infty$ algebras and higher geometric structures
  12. Homotopy Rota-Baxter operators and post-Lie$_\infty$ algebras
  13. Homotopy Rota-Baxter operators on open-closed homotopy Lie algebras
  14. Post-Lie$_\infty$ algebras and homotopy Rota-Baxter operators

Key Result

Proposition 2.10

With notations as above, $\Phi$ is an isomorphism from the graded vector space $\mathrm{Hom}(V,\mathsf{S}(V)\otimes \bar{\mathsf{S}}(V))$ to $\mathsf{Der}(\mathsf{S}(V)\otimes \bar{\mathsf{S}}(V))$.

Theorems & Definitions (78)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5: Free post-Lie algebra
  • Example 2.6: Ihara Lie algebra
  • Definition 2.7
  • Example 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 68 more