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

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); 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.

11 pages. This is a completely revised version of an earlier submission arXiv:1510.07544