Homotopy BV algebras in Poisson geometry

TL;DR

This paper introduces the degeneration property for BV-infinity algebras, demonstrating its implications for homotopy abelian L-infinity algebras and showing that higher Koszul brackets vanish in certain Poisson structures.

Contribution

It generalizes a key identity in BV algebras and applies it to prove the vanishing of higher Koszul brackets in generalized Poisson structures.

Findings

Degeneration property implies homotopy abelian L-infinity algebras.

Higher Koszul brackets vanish on cohomology with generalized Poisson structures.

Generalization of the identity (e^x)=e^x(\u0014(x)+[x,x]/2) in BV algebras.

Abstract

We define and study the degeneration property for BV-infinity algebras and show that it implies that the underlying L-infinity algebras are homotopy abelian. The proof is based on a generalisation of the well-known identity \Delta(e^x)=e^x(\Delta(x)+[x,x]/2) which holds in all BV algebras. As an application we show that the higher Koszul brackets on the cohomology of a manifold supplied with a generalised Poisson structure all vanish.