Kirillov structures up to homotopy

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∞-algebra, which we refer to as a homotopy Kirillov algebra. We are then led 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.MSC (2010):16E45, 53D17, 58A50, 58E40.

The Local Structure of Generalized Contact Bundles

“…In principle, applying the dictionary is straightforward: it is enough to replace functions on a manifold M with sections of a line bundle L → M , vector fields over M with derivations of L, etc. In practice, applying the dictionary can be actually challenging, and may lead to interesting new features [19,36,37,24,25,9,34,33,39,40].…”

Remarks on Contact and Jacobi Geometry

“…The picture of Jacobi geometry -or better Kirillov geometry -in terms of R ×bundles and homogeneous Poisson structures allows for a natural generalisation thereof to the world of L ∞ -algebras via replacing the Poisson structure with a higher Poisson structure (also known as a P ∞ -structure). This leads to the notions of higher Kirillov manifolds and Kirillov structures up to homotopy, see [6].…”