Bracket-Preserving property of Anchor Maps and Applications to Various Brackets

TL;DR

This paper proves that the anchor map preserves brackets in certain algebraic structures, simplifying their definitions and applying this to various brackets, including those in Courant algebroids and Nambu-Poisson manifolds.

Contribution

It demonstrates the bracket-preserving property of anchor maps, revealing redundancy in the homomorphism condition of Leibniz algebroids and applying this to multiple bracket structures.

Findings

Anchor map preserves brackets in Leibniz algebroids.

Redundancy of the homomorphism condition in Leibniz algebroids.

Application to brackets in Courant algebroids and 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); an aspect not addressed in the literature. We apply our result to the brackets of Hagiwara, Ibanez et. al; we settle an old query of Uchino on redundancy for Courant bracket.