Kirillov structures up to homotopy

TL;DR

This paper introduces higher Kirillov brackets and algebroids on supermanifolds, extending classical Jacobi structures to a homotopy framework and establishing their algebraic properties and relations.

Contribution

It develops the theory of homotopy Kirillov structures, generalizing Jacobi algebroids and associating them with homotopy BV-algebras.

Findings

Higher Kirillov brackets form an $L_{}$-algebra structure.

Construction of higher Kirillov algebroids as generalizations of Jacobi algebroids.

Establishment of a correspondence between higher Kirillov manifolds and homotopy BV-algebras.

Abstract

We present the notion of higher Kirillov brackets on the sections of an even line bundle over a supermanifold. When the line bundle is trivial we shall speak of higher Jacobi brackets. These brackets are understood furnishing the module of sections with an $L_{\infty}$-algebra, which we refer to as a homotopy Kirillov algebra. We are then to higher Kirillov algebroids as higher generalisations of Jacobi algebroids. Furthermore, we show how to associate a higher Kirillov algebroid and a homotopy BV-algebra with every higher Kirillov manifold. In short, we construct homotopy versions of some of the well-known theorems related to Kirillov's local Lie algebras.