Jacobi structures in supergeometric formalism
Paulo dos Santos Antunes, Camille Laurent-Gengoux

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.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Nonlinear Waves and Solitons
