The graded Jacobi algebras and (co)homology

Abstract

Jacobi algebroids (i.e. `Jacobi versions' of Lie algebroids) are studied in the context of graded Jacobi brackets on graded commutative algebras. This unifies varios concepts of graded Lie structures in geometry and physics. A method of describing such structures by classical Lie algebroids via certain gauging (in the spirit of E.Witten's gauging of exterior derivative) is developed. One constructs a corresponding Cartan differential calculus (graded commutative one) in a natural manner. This, in turn, gives canonical generating operators for triangular Jacobi algebroids. One gets, in particular, the Lichnerowicz-Jacobi homology operators associated with classical Jacobi structures. Courant-Jacobi brackets are obtained in a similar way and use to define an abstract notion of a Courant-Jacobi algebroid and Dirac-Jacobi structure. All this offers a new flavour in understanding the Batalin-Vilkovisky formalism.

20 pages, a few typos corrected; final version to be published in J. Phys. A: Math. Gen

References in corpus (3)

• Jacobi structures revisited
• Conformal Dirac Structures
• Lie algebroid foliations and ${\mathcal{E}}^1(M)$-Dirac structures

Cited by in corpus (25)

• Higher vector bundles and multi-graded symplectic manifolds
• Graded contact manifolds and contact Courant algebroids
• Contact manifolds and generalized complex structures
• AV-differential geometry: Poisson and Jacobi structures
• A novel approach to contact Hamiltonians and contact Hamilton-Jacobi theory
• Remarks on Contact and Jacobi Geometry
• Courant algebroid and Lie bialgebroid contractions
• Brackets
• Courant Algebroids. A Short History
• Integration of Dirac-Jacobi structures
• Twisted Jacobi manifolds, twisted Dirac-Jacobi structures and quasi-Jacobi bialgebroids
• E-Courant algebroids
• Generalized Contact Geometry and T-Duality
• Poisson-Jacobi reduction of homogeneous tensors
• Reductions: precontact versus presymplectic
• VB-structures and generalizations
• Jacobi-Nijenhuis algebroids and their modular classes
• On Jacobi quasi-Nijenhuis algebroids and Courant-Jacobi algebroid morphisms
• On quasi-Jacobi and Jacobi-quasi bialgebroids
• Jacobi quasi-Nijenhuis Algebroids
• The Schroedinger operator as a generalized Laplacian
• Hopf algebroids associated to Jacobi algebras
• Jacobi Geometry and Hamiltonian Mechanics: the Unit-Free Approach
• Mechanical Hamiltonian systems with respect to linear Poisson structures and Jacobi-Reeb dynamics
• Generating Operators of the Krasil'shchik-Schouten bracket