2015/04/03 by M. Jotz Lean, Lean, M. Jotz · 3 citations
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.1504.00880
openalex publication_date 2015/04/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
This paper shows the equivalence of the categories of N-manifolds of degree 2 with the category of double vector bundles endowed with a linear metric. Split Poisson N-manifolds of degree 2 are shown to be equivalent to self-dual representations up to homotopy. As a consequence, the equivalence above induces an equivalence between so called metric VB-algebroids and Poisson N-manifolds of degree 2. Then a new description of split Lie 2-algebroids is given, as well as their "duals", the Dorfman 2-representations. We show that Dorfman 2-representations are equivalent in a simple manner to Lagrangian splittings of VB-Courant algebroids. This yields the equivalence of the categories of Lie 2-algebroids and of VB-Courant algebroids. We give several natural classes of examples of split Lie 2-algebroids and of the corresponding VB-Courant algebroids. We then show that a split Poisson Lie 2-algebroid is equivalent to the "matched pair" of a Dorfman 2-representation with a self-dual representation up to homotopy. We deduce a new proof of the equivalence of categories of LA-Courant algebroids and Poisson Lie 2-algebroids. We show that the core of an LA-Courant algebroid inherits naturally the structure of a degenerate Courant algebroid. This yields a new formula to retrieve in a direct manner the Courant algebroid found by Roytenberg to correspond to a symplectic Lie 2-algebroid. Finally we study VB- and LA-Dirac structures in VB- and LA-Courant algebroids. As an application, we extend Li-Bland's results on pseudo-Dirac structures and we construct a Manin pair associated to an LA-Dirac structure.