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The geometric constraints on Filippov algebroids

Yanhui Bi, Zhixiong Chen, Zhuo Chen, Maosong Xiang

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.

The geometric constraints on Filippov algebroids

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.
Paper Structure (8 sections, 10 theorems, 59 equations)

This paper contains 8 sections, 10 theorems, 59 equations.

Table of Contents

  1. Preliminaries: Anchored bundles and Filippov algebroids
  2. The geometric constraints of Filippov algebroids
  3. The main theorem
  4. Proof of Theorem \ref{['th2.3']}
  5. Construction of Filippov connections
  6. A construction of linear Nambu-Poisson structures
  7. From Filippov algebroids to linear Nambu-Poisson structures
  8. Proof of Theorem \ref{['prop3.3']}

Key Result

Proposition 1.5

Let $(A,[\cdotp,\cdotp \cdotp \cdotp,\cdotp],\rho)$ be a Filippov $n$-algebroid for $n\geqslant 3$. Then the rank of the image of $\rho$ as a distribution on $M$ can not exceed $1$, i.e., $\operatorname{rank}(\rho(\wedge^{n-1}A))\leqslant 1$.

Theorems & Definitions (27)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Example 1.4
  • Proposition 1.5
  • proof
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Lemma 2.4
  • ...and 17 more