Homotopy derivations

TL;DR

This paper introduces the concept of strong homotopy derivations for operad-based algebras, describing their properties, symmetrization, and providing examples, thereby extending the understanding of derivations in homotopy algebra contexts.

Contribution

It defines strong homotopy derivations for operad algebras, describes their structure for Koszul operads, and explores their symmetrization and examples, advancing the theory of homotopy derivations.

Findings

Strong homotopy derivations are coderivations closed under Lie bracket.

Symmetrization of A-infinity derivations yields L-infinity derivations.

Examples include generalizations of inner derivations.

Abstract

We define a strong homotopy derivation of (cohomological) degree k of a strong homotopy algebra over an operad P. This involves resolving the operad obtained from P by adding a generator with "derivation relations". For a wide class of Koszul operads P, in particular Ass and Lie, we describe the strong homotopy derivations by coderivations and show that they are closed under the Lie bracket. We show that symmetrization of a strong homotopy derivation of an A-infinity algebra yields a strong homotopy derivation of the symmetrized L-infinity algebra. We give examples of strong homotopy derivations generalizing inner derivations.