Exploring the Structure of Higher Algebroids

TL;DR

This paper investigates the algebraic foundations of higher-order algebroids, generalizing Lie algebroids using a comorphism approach, and establishes a correspondence for the case k=2 involving representations and morphisms.

Contribution

It uncovers the algebraic structures underlying higher-order algebroids and provides a new correspondence for the case k=2 involving representations and morphisms.

Findings

Established a one-to-one correspondence for k=2 between higher-order Lie algebroids and pairs of representations and morphisms.

Uncovered the algebraic structures underlying higher-order algebroids through the comorphism framework.

Extended the understanding of higher algebroids in the context of geometric mechanics.

Abstract

The notion of a \emph{higher-order algebroid}, as introduced by J\'o\'zwikowski and Rotkiewicz in their work \emph{Higher-order analogs of Lie algebroids via vector bundle comorphisms} (SIGMA, 2018), generalizes the concepts of a higher-order tangent bundle $\tau^k_M: \mathrm{T}^k M \to M$ and a (Lie) algebroid. This idea is based on a (vector bundle) comorphism approach to (Lie) algebroids and the reduction procedure of homotopies from the level of Lie groupoids to that of Lie algebroids. In brief, an alternative description of a Lie algebroid $(A, [\cdot, \cdot], \sharp)$ is a vector bundle comorphism $\kappa$, defined as the dual of the Poisson map $\varepsilon: \mathrm{T}^\ast A \to \mathrm{T} A^\ast$ associated with the Lie algebroid $A$. The framework of comorphisms has proven to be a suitable language for describing higher-order analogues of Lie algebroids from the perspective of the role played by (Lie) algebroids in geometric mechanics. In this work, we uncover the classical algebraic structures underlying the somewhat mysterious description of higher-order algebroids through comorphisms. For the case $k=2$, we establish a one-to-one correspondence between higher-order Lie algebroids and pairs consisting of a two-term representation (up to homotopy) of a Lie algebroid and a morphism to the adjoint representation of this algebroid.