Kapranov DG-manifolds and Poincar{\'e}--Birkhoff--Witt isomorphisms
Résumé
We prove that to every pair of Lie algebroids (L, A) corresponds a Kapranov dg-manifold structure on A[1] ⊕ L/A, canonical up to isomorphism. As a consequence, Γ(∧ • A ∨ ⊗ L/A) carries a canonical L∞[1] algebra structure whose binary bracket is a cocycle representative of the Atiyah class of the pair (L, A). For Lie algebroids over R, we conjecture that this Kapranov dg-manifold ought to be considered as the derived formal neighborhood of a certain substack in a double-quotient differentiable stack. The second main purpose of the paper is the construction, for Lie algebroid pairs, of Poincaré– Birkhoff–Witt type isomorphisms described by an iteration formula, which allows explicit computations. Such Poincaré-Birkhoff-Witt isomorphisms constitute simultaneous extensions of the classical PBW map of Lie theory and the inverse of the complete symbol map of differential operators.
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)
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