$L_\infty$-interpretation of a classification of deformations of Poisson structures in dimension three

TL;DR

This paper provides an $L_$-algebra framework to interpret and explicitly classify formal deformations of certain exact Poisson structures in three dimensions, enhancing understanding of their deformation theory.

Contribution

It introduces an $L_$-interpretation and constructs explicit formulas and classifications for deformations of specific Poisson structures in dimension three.

Findings

Explicit formulas for all formal deformations obtained.

Classification of deformations in the generic case established.

A quasi-isomorphism between $L_$-algebras constructed to relate structures.

Abstract

We give an $L_\infty$-interpretation of the classification, obtained in [AP2], of the formal deformations of a family of exact Poisson structures in dimension three. We indeed obtain again the explicit formulas for all the formal deformations of these Poisson structures, together with a classification in the generic case, by constructing a suitable quasi-isomorphism between two $L_\infty$-algebras, which are associated to these Poisson structures.