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See detailAtiyah classes and dg-Lie algebroids for matched pairs
Batakidis, Panagiotis; Voglaire, Yannick UL

in Journal of Geometry & Physics (2017)

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 ... [more ▼]

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. [less ▲]

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See detailInvariant connections and PBW theorem for Lie groupoid pairs
Laurent-Gengoux, Camille; Voglaire, Yannick UL

E-print/Working paper (2015)

Given a closed wide Lie subgroupoid $\mathbf{A}$ of a Lie groupoid $\mathbf{L}$, i.e. a Lie groupoid pair, we interpret the associated Atiyah class as the obstruction to the existence of $\mathbf{L ... [more ▼]

Given a closed wide Lie subgroupoid $\mathbf{A}$ of a Lie groupoid $\mathbf{L}$, i.e. a Lie groupoid pair, we interpret the associated Atiyah class as the obstruction to the existence of $\mathbf{L}$-invariant fibrewise affine connections on the homogeneous space $\mathbf{L}/\mathbf{A}$. For Lie groupoid pairs with vanishing Atiyah class, we show that the left $\mathbf{A}$-action on the quotient space $\mathbf{L}/\mathbf{A}$ can be linearized. In addition to giving an alternative proof of a result of Calaque about the Poincare-Birkhoff-Witt map for Lie algebroid pairs with vanishing Atiyah class, this result specializes to a necessary and sufficient condition for the linearization of dressing actions, and gives a clear interpretation of the Molino class as an obstruction to simultaneous linearization of all the monodromies. In the course of the paper, a general theory of connections and connection forms on Lie groupoid principal bundles is developed. Also, a computational substitute to the adjoint action (which only exists "up to homotopy") is suggested. [less ▲]

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See detailRozansky-Witten-type invariants from symplectic Lie pairs
Voglaire, Yannick UL; Xu, Ping

in Communications in Mathematical Physics (2014)

We introduce symplectic structures on “Lie pairs” of (real or complex) Lie algebroids as studied by Chen et al. (From Atiyah classes to homotopy Leibniz algebras. arXiv:1204.1075, 2012), encompassing ... [more ▼]

We introduce symplectic structures on “Lie pairs” of (real or complex) Lie algebroids as studied by Chen et al. (From Atiyah classes to homotopy Leibniz algebras. arXiv:1204.1075, 2012), encompassing homogeneous symplectic spaces, symplectic manifolds with a g-action, and holomorphic symplectic manifolds. We show that to each such symplectic Lie pair are associated Rozansky–Witten-type invariants of three-manifolds and knots, given respectively by weight systems on trivalent and chord diagrams. [less ▲]

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See detailStrongly exponential symmetric spaces
Voglaire, Yannick UL

in International Mathematics Research Notices (2013)

We study the exponential map of connected symmetric spaces and characterize, in terms of midpoints and of infinitesimal conditions, when it is a diffeomorphism, generalizing the Dixmier–Saito theorem for ... [more ▼]

We study the exponential map of connected symmetric spaces and characterize, in terms of midpoints and of infinitesimal conditions, when it is a diffeomorphism, generalizing the Dixmier–Saito theorem for solvable Lie groups. We then give a geometric characterization of the (strongly) exponential solvable symmetric spaces as those spaces for which every triangle admits of a unique double triangle. This work is motivated by Weinstein's quantization by groupoids program applied to symmetric spaces. [less ▲]

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See detailQuantized Anti de Sitter spaces and non-formal deformation quantizations of symplectic symmetric spaces
Bieliavsky, Pierre; Claessens, Laurent; Sternheimer, Daniel et al

in Dito, Giuseppe; Lu, Jiang-Hua; Maeda, Yoshiaki (Eds.) et al Poisson Geometry in Mathematics and Physics (2008)

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