2024/08/05 by Mikołaj Rotkiewicz, Rotkiewicz, Mikołaj
Computer Science · Mathematics · #17B66 #17B70 #58A20 #58A50 #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Polynomial and algebraic computation #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2408.02194
openalex publication_date 2024/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The notion of a higher-order algebroid, as introduced by Jóźwikowski and Rotkiewicz in their work Higher-order analogs of Lie algebroids via vector bundle comorphisms (SIGMA, 2018), generalizes the concepts of a higher-order tangent bundle τkM: Tk M → 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, [⋅, ⋅], \sharp) is a vector bundle comorphism κ, defined as the dual of the Poisson map ε: T^∗ A → T A^∗ 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.