On the axioms of Leibniz algebroids associated to Nambu-Poisson manifolds

Abstract

Let $E\to M$ be a smooth vector bundle with a bilinear product on $\Gamma (E)$ satisfying the Jacobi identity. Assuming only the existence of an anchor map $\mathfrak{a}$ we show that $\mathfrak{a}([X,Y])=[\mathfrak{a}X,\mathfrak{a}Y{]}_c$. This gives the redundancy of the homomorphism condition in the definition of Leibniz algebroid; in particular if it arises from a Nambu-Poisson manifold.

arXiv comment5 pages