Odd Jacobi manifolds and Loday-Poisson brackets

In this paper we construct a non-skewsymmetric version of a Poisson bracket on the algebra of smooth functions on an odd Jacobi supermanifold. We refer to such Poisson-like brackets as Loday-Poisson brackets. We examine the relations between the Hamiltonian vector fields with respect to both the odd Jacobi structure and the Loday-Poisson structure. Furthermore, we show that the Loday-Poisson bracket satisfies the Leibniz rule over the noncommutative product derived from the homological vector field.Comment: 18 pages. Further comments added and slight rearrangement of the material. No major changes to the mathematical content. Comments welcome

Modular classes of Q-manifolds: a review and some applications

A Q-manifold is a supermanifold equipped with an odd vector field that squares to zero. The notion of the modular class of a Q-manifold -- which is viewed as the obstruction to the existence of a Q-invariant Berezin volume -- is not well know. We review the basic ideas and then apply this technology to various examples, including $L_{\infty}$-algebroids and higher Poisson manifolds.Comment: 11 pages. Comments are welcome

Odd Jacobi manifolds: general theory and applications to generalised Lie algebroids

In this paper we define a Grassmann odd analogue of Jacobi structure on a supermanifold. The basic properties are explored. The construction of odd Jacobi manifolds is then used to reexamine the notion of a Jacobi algebroid. It is shown that Jacobi algebroids can be understood in terms of a kind of curved Q-manifold, which we will refer to as a quasi Q-manifold.Comment: 35 pages. This preprint is an amalgamation of the earlier preprints arXiv:111.4044, arXiv:1103.1803 and arXiv:1101.1844. A version of this work appears in Extracta Mathematica