On the Lie algebroid of a derived self-intersection

Calaque, Damien; Căldăraru, Andrei; Tu, Junwu

doi:10.1016/j.aim.2014.06.002

Cited by 18 publications

(25 citation statements)

References 21 publications

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“…We want to mention that Calaque, Cȃldȃraru and Tu have established similar results in the algebraic setting ( [CCT14]). While they built a dg-Lie algebroid on some dg-sheaf which is quasi-isomorphic to i * N[−1] in the derived category of Y, our L ∞ -algebroid has the Dolbeault resolution of the normal bundle as the underlying complex and the higher brackets do not vanish in general.…”

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confidence: 54%

Dolbeault dga and L∞-algebroid of the formal neighborhood

Shi

2017

Advances in Mathematics

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We continue the study the Dolbeault dga of the formal neighborhood of an arbitary closed embedding of complex manifolds previously defined by the author in [Yua]. The special case of the diagonal embedding has been studied in [Yu15]. We describe the Dolbeault dga explicitly in terms of the formal differential geometry of the embedding. Moreover, we show that the Dolbeault dga is the completed Chevalley-Eilenberg dga an L ∞ -algebroid structure on the shifted normal bundle of the submanifold. This generlizes the result of Kapranov on the diagonal embedding and Atiyah class.

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confidence: 54%

Dolbeault dga and L∞-algebroid of the formal neighborhood

Shi

2017

Advances in Mathematics

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“…He showed that there was an adjunction between X-pointed, X-linear formal moduli problems and dg Lie algebras in QCoh(X). (3) In the setting of smooth algebraic varieties in characteristic zero, Calaque-Cȃldȃraru-Tu [CCT14] gave an adjunction between X-pointed, K-linear formal moduli problems and Lie algebroids on X. (4) Gaitsgory-Rozenblyum [GR17] greatly generalize the notion of Lie algebroid and formal moduli problems, to a theory internal to derived stacks (and phrased in terms of (∞, 2)categories).…”

Section: 4mentioning

confidence: 99%

“…We hope this pattern continues. (We wonder, in particular, about L ∞ space analogs of, e.g., the work of Calaque, Cȃldȃraru, and Tu [CCT14] on Lie algebroids for derived intersections.) 1.2.…”

Section: Introductionmentioning

confidence: 99%

Lie Algebroids as Spaces

Grady

Gwilliam

2018

J. Inst. Math. Jussieu

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In this paper, we relate Lie algebroids to Costello's version of derived geometry. For instance, we show that each Lie algebroid-and the natural generalization to dg Lie algebroidsprovides an (essentially unique) L∞ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of L∞ spaces. Then we show that for each Lie algebroid L, there is a fully faithful functor from the category of representations up to homotopy of L to the category of vector bundles over the associated L∞ space. Indeed, this functor sends the adjoint complex of L to the tangent bundle of the L∞ space. Finally, we show that a shifted-symplectic structure on a dg Lie algebroid produces a shifted symplectic structure on the associated L∞ space.

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“…X can be quantized as h-modules X is a tame quantized cycle [π h ([m, m]), m] = 0 X admits a retraction at the second order in Y g = h ⋊ m If one of the two last conditions is satisfied, the object N X/Y [−1] is naturally a Lie object in D b (X), but this is no longer the case if we drop the tameness assumption. In full generality (that is without any specific quantization conditions), the algebraic structure of N X/Y [−1] has been investigated in [5]: it is a derived Lie algebroid, whose anchor map is given by the extension class of the normal exact sequence of the pair (X, Y). Hence, our setting can be understood as the weaker universal hypotheses for which this derived Lie algebroid is a true Lie object in the symmetric monoidal category D b (X).…”

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confidence: 99%

“…The main subtlety here lies in the fact that if the Lie algebroid structure of N X/Y [−1] in in fact a true Lie structure in D b (X), the universal envelopping algebras of N X/Y [−1] as a Lie algebra object or as a derived Lie algebroid are not the same; they don't even live in the same categories. In the setting of [5],…”

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confidence: 99%

The Ext algebra of a quantized cycle

Calaque¹,

Grivaux²

2019

Journal De L’École Polytechnique — Mathématiques

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Given a quantized analytic cycle (X, σ) in Y, we give a categorical Lie-theoretic interpretation of a geometric condition, discovered by Shilin Yu, that involves the second formal neighbourhood of X in Y. If this condition (that we call tameness) is satisfied, we prove that the derived Ext algebrais isomorphic to the universal enveloping algebra of the shifted normal bundle N X/Y [−1] endowed with a specific Lie structure, strengthening an earlier result of Cȃldȃraru, Tu, and the first author This approach allows to get some conceptual proofs of many important results in the theory: in the case of the diagonal embedding, we recover former results of Kapranov, Markarian, and Ramadoss about (a) the Lie structure on the shifted tangent bundle T X [−1] (b) the corresponding universal enveloping algebra (c) the calculation of Kapranov's big Chern classes. We also give a new Lie-theoretic proof of Yu's result for the explicit calculation of the quantized cycle class in the tame case: it is the Duflo element of the Lie algebra object N X/Y [−1].

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On the Lie algebroid of a derived self-intersection

Cited by 18 publications

References 21 publications

Dolbeault dga and L∞-algebroid of the formal neighborhood

Dolbeault dga and L∞-algebroid of the formal neighborhood

Lie Algebroids as Spaces

The Ext algebra of a quantized cycle

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