Matched pairs of Leibniz algebroids, Nambu–Jacobi structures and modular class

Abstract

The notion of a matched pair of Leibniz algebroids is introduced and it is shown that a Nambu–Jacobi structure of order n, n>2, over a manifold M defines a matched pair of Leibniz algebroids. As a consequence, one deduces that the vector bundle $\text{⋀}{}^{\mathrm{n-1}}\text{(}\text{T}{}^{\mathrm{\ast }}\text{M)⊕⋀}{}^{\mathrm{n-2}}\text{(}\text{T}{}^{\mathrm{\ast }}\text{M)→M}$ is a Leibniz algebroid. Finally, if M is orientable, the modular class of M is defined as a cohomology class of order 1 with respect to this Leibniz algebroid.