The geometric constraints on Filippov algebroids

Abstract

<abstract><p>Filippov $ n $-algebroids are introduced by Grabowski and Marmo as a natural generalization of Lie algebroids. On this note, we characterized Filippov $ n $-algebroid structures by considering certain multi-input connections, which we called Filippov connections, on the underlying vector bundle. Through this approach, we could express the $ n $-ary bracket of any Filippov $ n $-algebroid using a torsion-free type formula. Additionally, we transformed the generalized Jacobi identity of the Filippov $ n $-algebroid into the Bianchi-Filippov identity. Furthermore, in the case of rank $ n $ vector bundles, we provided a characterization of linear Nambu-Poisson structures using Filippov connections.</p></abstract>