2010/01/31 by Olivier Brahic, O. Brahic, Chenchang Zhu · 1 citation
Mathematics · Physics and Astronomy · #Adjoint representation of a Lie algebra #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Connection (principal bundle) #Degree (music) #Exact sequence #Fibration #Geometry #Homotopy #Homotopy and Cohomology in Algebraic Topology #Lie algebra #Lie algebroid #Lie bracket of vector fields #Manifold (fluid mechanics) #Mathematics #Pure mathematics #Vector field #Weight #math-ph #math.DG #math.MP #msc:14F35 #msc:22A22 #msc:53D17
paper · pdf · doi:10.1016/j.aim.2010.10.006
28 pages, 1 figure
arxiv created 2010/02/01 · openalex publication_date 2010/10/28 · arxiv updated 2011/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
A degree 1 non-negative graded super manifold equipped with a degree 1 vector field Q satisfying [Q, Q]=1, namely a so-called NQ-1 manifold is, in plain differential geometry language, a Lie algebroid. We introduce a notion of fibration for such super manifols, that essentially involves a complete Ehresmann connection. As it is the case for Lie algebras, such fibrations turn out not to be just locally trivial products. We also define homotopy groups and prove the expected long exact sequence associated to a fibration. In particular, Crainic and Fernandes's obstruction to the integrability of Lie algebroids is interpreted as the image of a transgression map in this long exact sequence.