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Higher order Massey products for algebras over algebraic operads

Oisín Flynn-Connolly, José M. Moreno-Fernández

Abstract

We introduce higher-order Massey products for algebras over algebraic operads. This extends the work of Fernando Muro on secondary ones. We study their basic properties and behavior with respect to morphisms of algebras and operads and give some connections to formality. We prove that these higher-order operations represent the differentials in a naturally associated operadic Eilenberg--Moore spectral sequence. We also study the interplay between particular choices of higher-order Massey products and quasi-isomorphic $\mathcal P_\infty$-structures on the homology of a $\mathcal P$-algebra. We focus on Koszul operads over a characteristic zero field and explain how our results generalize to the non-Koszul case.

Higher order Massey products for algebras over algebraic operads

Abstract

We introduce higher-order Massey products for algebras over algebraic operads. This extends the work of Fernando Muro on secondary ones. We study their basic properties and behavior with respect to morphisms of algebras and operads and give some connections to formality. We prove that these higher-order operations represent the differentials in a naturally associated operadic Eilenberg--Moore spectral sequence. We also study the interplay between particular choices of higher-order Massey products and quasi-isomorphic -structures on the homology of a -algebra. We focus on Koszul operads over a characteristic zero field and explain how our results generalize to the non-Koszul case.
Paper Structure (10 sections, 17 theorems, 168 equations, 2 figures)

This paper contains 10 sections, 17 theorems, 168 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Operadic background
  4. Higher-order Massey products for associative algebras
  5. Secondary Massey products for algebras over algebraic operads
  6. Higher-order operadic Massey products
  7. Elementary properties of the operadic Massey products
  8. Massey products along morphisms of operads and formality
  9. Differentials in the $\mathcal{P}$-Eilenberg--Moore spectral sequence
  10. Higher-order operadic Massey products and $\mathcal{P}_\infty$-structures

Key Result

Theorem 1.2

Let $(A, d)$ be a $\mathcal{P}$-algebra, $H$ its homology, and $f : H \to A$ a cycle-choosing (and therefore necessarily degree $0$) linear map. Let $\delta_A$ be the degree $-1$ square-zero coderivation of $\mathcal{P}^{\text{\textexclamdown}}(A)$ representing the $\mathcal{P}$-algebra structure on

Figures (2)

  • Figure 1: The Massey inductive map for $\mathsf{Ass}$
  • Figure 2: The Massey inductive map for $\mathsf{Lie}$

Theorems & Definitions (52)

  • Theorem 1.2
  • proof : Sketch of the proof
  • Definition 1.3
  • Definition 1.4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • ...and 42 more