Reference : Atiyah classes and dg-Lie algebroids for matched pairs
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
Atiyah classes and dg-Lie algebroids for matched pairs
Batakidis, Panagiotis []
Voglaire, Yannick mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Journal of Geometry & Physics
Elsevier Science
Yes (verified by ORBilu)
The Netherlands
[en] differential graded manifolds ; Atiyah classes ; Lie algebroids ; Fedosov resolutions
[en] For every Lie pair $(L,A)$ of algebroids we construct a dg-manifold structure on the $\ZZ$-graded manifold $\M=L[1]\oplus L/A$ such that the inclusion $\iota: A[1] \to \M$ and the projection $p:\M\to L[1]$ are morphisms of dg-manifolds.
The vertical tangent bundle $T^p\M$ then inherits a structure of dg-Lie algebroid over $\M$.
When the Lie pair comes from a matched pair of Lie algebroids, we show that the inclusion $\iota$ induces a quasi-isomorphism that sends the Atiyah class of this dg-Lie algebroid to the Atiyah class of the Lie pair.
We also show how (Atiyah classes of) Lie pairs and dg-Lie algebroids give rise to (Atiyah classes of) dDG-algebras.

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