The geometric constraints on Filippov algebroids

TL;DR

This paper explores the geometric structure of Filippov n-algebroids, introducing Filippov connections to express their brackets and identities, and characterizes linear Nambu-Poisson structures in this framework.

Contribution

It introduces Filippov connections to characterize Filippov n-algebroids and relates their identities to classical geometric identities, extending the understanding of these structures.

Findings

Expressed n-ary brackets via torsion-free formulas

Reformulated the Jacobi identity as Bianchi-Filippov identity

Characterized linear Nambu-Poisson structures using Filippov connections

Abstract

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.