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arXiv · 1607.00807

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

S. Srinivas Rau·T. Shreecharan
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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.

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S. Srinivas Rau, T. Shreecharan. 2016-07-04. Bracket-Preserving property of Anchor Maps and Applications to Various Brackets. https://arxiv.org/abs/1607.00807

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