Jacobi structures in supergeometric formalism

TL;DR

This paper employs supergeometric formalism to provide a unified and simplified framework for understanding Jacobi algebroids and bialgebroids, including their structures, Poissonization, and compatibility conditions.

Contribution

It introduces a supergeometric approach to define and analyze Jacobi algebroids and bialgebroids, simplifying their descriptions and interrelations.

Findings

Supergeometric formalism efficiently encodes brackets and anchors.

Defines Jacobi-Gerstenhaber algebra structure for Jacobi algebroids.

Provides a simple description of Jacobi bialgebroids and their relation to Jacobi structures.

Abstract

We use the supergeometric formalism, more precisely, the so-called "big bracket" (for which brackets and anchors are encoded by functions on some graded symplectic manifold) to address the theory of Jacobi algebroids and bialgebroids (following mainly Iglesias-Marrero and Grabowski-Marmo as a guideline). This formalism is in particular efficient to define the Jacobi-Gerstenhaber algebra structure associated to a Jacobi algebroid, to define its Poissonization, and to express the compatibility condition defining Jacobi bialgebroids. Also, we claim that this supergeometric language gives a simple description of the Jacobi bialgebroid associated to Jacobi structures, and conversely, of the Jacobi structure associated to Jacobi bialgebroid.