Atiyah classes of Lie algebroid homotopy modules

2024/06/07 by Panagiotis Batakidis, Batakidis, Panagiotis, Sylvain Lavau +1
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph)

paper · pdf · doi:10.48550/arxiv.2406.05204

openalex publication_date 2024/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a Lie algebroid pair A\hookrightarrow L we study cocycles constructed from the extension to L of the higher connection forms of a representation up to homotopy E of the Lie algebroid A. We show that there exists a cohomology class with values in the endomorphism bundle of E that is independent of the extension above and vanishes whenever a homotopy A-compatible extension exists. Whenever the representation up to homotopy E is the resolution of a Lie algebroid representation K of A, it is shown that there exists a quasi-isomorphism sending the new Atiyah class to the classical one, associated to extensions to L of the Lie algebroid representation K.

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