The graded Jacobi algebras and (co)homology

Jacobi algebroids (i.e. 'Jacobi versions' of Lie algebroids) are studied in the context of graded Jacobi brackets on graded commutative algebras. This unifies various 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 the exterior derivative) is developed. One constructs a corresponding Cartan differential calculus (graded commutative 0 e) 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 used to define an abstract notion of a Courant-Jacobi algebroid and Dirac-Jacobi structure.