Matched pairs of Leibniz algebroids, Nambu-Jacobi structures and modular class (Q2765873)

The authors introduce the notion of a matched pair of Leibniz algebroids. Let us recall that a Leibniz algebroid is a noncommutative version of a Lie algebroid. An important example is the bundle \(\Lambda^{n-1}T^*M \rightarrow M\) of \((n-1)\)-forms on a Nambu-Poisson manifold \(M\) of order \(n\). A Nambu-Poisson manifold of order \(n\) is a manifold \(M\) equipped with an \(n\)-vector \(\Lambda\) (equivalently, with a skew-symmetric bracket of \(n\) functions which acts as a derivation on each of its arguments) satisfying some integrability conditions.NEWLINENEWLINENEWLINEGiven two Leibniz algebroids \({\mathcal A}_1\) and \({\mathcal A}_2\) over the same basis, we say that they form a matched pair if the Whitney sum \({\mathcal A}_1 \oplus {\mathcal A}_2\) is again a Leibniz algebroid containing \({\mathcal A}_1\) and \({\mathcal A}_2\) as Leibniz subalgebroids. On the other hand, a Nambu-Jacobi manifold of order \(n\) is a manifold \(M\) equipped with an \(n\)-vector \(\Lambda\) and an \((n-1)\)-vector \(\square\) (equivalently, with a skew-symmetric bracket of \(n\) functions which acts as a first-order differential operator on each of its arguments) satisfying some integrability conditions. NEWLINENEWLINENEWLINEIt is proved that a Nambu-Jacobi manifold \((M, \Lambda, \square)\) of order \(n\) possesses a canonically associated matched pair of Leibniz algebroids \(\Lambda^{n-1}T^*M \rightarrow M\) and \(\Lambda^{n-2}T^*M \rightarrow M\). The modular class of the Nambu-Jacobi structure is just defined as a cohomology class in the cohomology associated to the Leibniz algebroid \(\Lambda^{n-1}T^*M \oplus \Lambda^{n-2}T^*M \rightarrow M\).