Higher order Massey products for algebras over algebraic operads

TL;DR

This paper introduces higher-order Massey products for algebras over algebraic operads, extending previous work, and explores their properties, connections to formality, and role in spectral sequences.

Contribution

It develops a theory of higher-order Massey products for operad algebras, analyzing their properties, relation to formality, and their role in spectral sequences, generalizing to non-Koszul operads.

Findings

Higher-order Massey products represent differentials in an operadic Eilenberg--Moore spectral sequence.

These products relate to quasi-isomorphic $ ext{P}_ ext{infty}$-structures on homology.

Results extend to non-Koszul operads over characteristic zero fields.

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.