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

TL;DR

This paper investigates the foundational axioms of Leibniz algebroids linked to Nambu-Poisson manifolds, revealing redundancies in their defining conditions and clarifying their structural properties.

Contribution

It demonstrates that the homomorphism condition in Leibniz algebroids is redundant under certain assumptions, especially for those arising from Nambu-Poisson manifolds.

Findings

The anchor map satisfies a compatibility condition with the bracket.

Redundancy of the homomorphism condition in Leibniz algebroid definitions.

Clarification of the structure of Leibniz algebroids associated with Nambu-Poisson manifolds.

Abstract

Let $E \rightarrow M$ 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.