Atiyah classes and dg-Lie algebroids for matched pairs

Abstract

For every Lie pair $(L,A)$ of algebroids we construct a dg-manifold structure on the $\mathbb{Z}$-graded manifold $\mathcal{M}=L[1]\oplus L/A$ such that the inclusion $\iota :A[1]\to \mathcal{M}$ and the projection $p:\mathcal{M}\to L[1]$ are morphisms of dg-manifolds. The vertical tangent bundle $T^p\mathcal{M}$ then inherits a structure of dg-Lie algebroid over $\mathcal{M}$. 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.

22 pages, to appear in J. Geom. Phys

References in corpus (2)

• The Atiyah class of a dg-vector bundle
• ${\mathbb{Z}}_2^n$-Supergeometry I: Manifolds and Morphisms

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• Hopf algebras arising from dg manifolds
• Polyvector fields and polydifferential operators associated with Lie pairs
• Internal symmetry of the $L_{⩽3}$ algebra arising from a Lie pair