<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="sans-serif">PBW</mml:mi></mml:math>for an inclusion of Lie algebras

Calaque, Damien; Căldăraru, Andrei; Tu, Junwu

doi:10.1016/j.jalgebra.2012.12.008

Cited by 15 publications

(34 citation statements)

References 7 publications

(50 reference statements)

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“…This corollary might be more relevant when seen at the level of jets where it immediately implies the following result, which gives back Theorem 1.5 of Calaque, Cȃldȃraru and Tu [10] for Lie algebras over R. (1) jets at eH of (k, l)-tensors on G/H, and (2) elements in S((g/h) * ) ⊗ k (g/h) * ⊗ l (g/h).…”

Section: Examples and Applicationsmentioning

confidence: 57%

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Invariant connections and PBW theorem for Lie groupoid pairs

Laurent-Gengoux

Voglaire

2019

Pacific J. Math.

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Given a closed wide Lie subgroupoid A of a Lie groupoid L, i.e. a Lie groupoid pair, we interpret the associated Atiyah class as the obstruction to the existence of L-invariant fibrewise affine connections on the homogeneous space L/A. For Lie groupoid pairs with vanishing Atiyah class, we show that the left A-action on the quotient space L/A can be linearized.In addition to giving an alternative proof of a result of Calaque about the Poincaré-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 on Lie groupoid equivariant principal bundles is developed. Contents2010 Mathematics Subject Classification. 53C05,53C30,53C12.

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Section: Examples and Applicationsmentioning

confidence: 57%

“…We list here some properties of κ that follow in a straightforward manner from the definitions. Below, L(·) and R(·) are the left-and right-invariant vector fields defined in (10). Proposition 3.11.…”

Section: An Adjoint Actionmentioning

confidence: 99%

Invariant connections and PBW theorem for Lie groupoid pairs

Laurent-Gengoux

Voglaire

2019

Pacific J. Math.

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“…Calaque and Van den Bergh [10] considered the Atiyah class of a DG module over a DG-algebra. They also inferred that, given a Lie algebra pair (d, g), the Atiyah class of the quotient d/g coincides with the class capturing the obstruction to the "PBW problem" studied earlier by Calaque-Cȃldȃraru-Tu [9] (see also [8,14]).…”

Section: Introductionmentioning

confidence: 64%

Atiyah Classes of Strongly Homotopy Lie Pairs

2019

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The subject of this paper is strongly homotopy (SH) Lie algebras, also known as L∞-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an SH Lie algebra A when it is extended to L. In fact, given such an SH Lie pair (L, A), and any A-module E, there associates a canonical cohomology class, the Atiyah class [α E ], which generalizes earlier known Atiyah classes out of Lie algebra pairs. We show that the Atiyah class [α L/A ] induces a graded Lie algebra structure on H • CE (A, L/A[−2]), and the Atiyah class [α E ] of any A-module E induces a Lie algebra module structure on H • CE (A, E). Moreover, Atiyah classes are invariant under gauge equivalent A-compatible infinitesimal deformations of L.

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“…We suspect, based on this, that there may exist a deep link between solutions of the generalized Kashiwara-Vergne problem [1] and the above isomorphisms between harmonic and Hochschild structures on an algebraic variety. It is likely that this link can be found using the ideas developed in [10], [11], [14], [37], and [40].…”

Section: Remark 93mentioning

confidence: 99%

Kontsevich's graph complex, GRT, and the deformation complex of the sheaf of polyvector fields

2015

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Kontsevich's graph complex, GRT, and the deformation complex of the sheaf of polyvector fields V.A. Dolgushev, C.L. Rogers, and T.H. WillwacherTo the memory of Boris Vasilievich Fedosov AbstractWe generalize Kontsevich's construction [38] of L ∞ -derivations of polyvector fields from the affine space to an arbitrary smooth algebraic variety. More precisely, we construct a map (in the homotopy category) from Kontsevich's graph complex to the deformation complex of the sheaf of polyvector fields on a smooth algebraic variety. We show that the action of Deligne-Drinfeld elements of the Grothendieck-Teichmüller Lie algebra on the cohomology of the sheaf of polyvector fields coincides with the action of odd components of the Chern character. Using this result, we deduce that theÂ-genus in the Calaque-Van den Bergh formula [12] for the isomorphism between harmonic and Hochschild structures can be replaced by a generalizedÂ-genus.

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PBWfor an inclusion of Lie algebras

Cited by 15 publications

References 7 publications

Invariant connections and PBW theorem for Lie groupoid pairs

Invariant connections and PBW theorem for Lie groupoid pairs

Atiyah Classes of Strongly Homotopy Lie Pairs

Kontsevich's graph complex, GRT, and the deformation complex of the sheaf of polyvector fields

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