Pre-Calabi--Yau algebras and homotopy double Poisson gebras

TL;DR

This paper establishes an equivalence between curved pre-Calabi--Yau algebras and curved homotopy double Poisson gerbes, linking two frameworks for derived noncommutative Poisson structures and exploring their homotopical properties.

Contribution

It proves the equivalence of two definitions of derived noncommutative Poisson structures and applies properadic calculus to analyze their homotopical features.

Findings

Equivalence between curved pre-Calabi--Yau algebras and curved homotopy double Poisson gerbes.

Isomorphism of the controlling differential graded Lie algebras for deformation theories.

Establishment of homotopical properties such as infinity-morphisms and formality.

Abstract

We prove that the notion of a curved pre-Calabi--Yau algebra is equivalent to the notion of a curved homotopy double Poisson gebra, thereby settling the equivalence between the two ways to define derived noncommutative Poisson structures. We actually prove that the respective differential graded Lie algebras controlling both deformation theories are isomorphic.This allows us to apply the recent developments of the properadic calculus in order to establish the homotopical properties of curved pre-Calabi--Yau algebras: infinity-morphisms, homotopy transfer theorem, formality, Koszul hierarchy, and twisting procedure.