Shifted Derived Poisson Manifolds Associated with Lie Pairs

Bandiera, Ruggero; Chen, Zhuo; Stiénon, Mathieu; Xu, Ping

doi:10.1007/s00220-019-03457-w

Cited by 19 publications

(51 citation statements)

References 48 publications

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“…The L ∞ -brackets {λ k } k≥5 vanish for similar reasons.It thus follows that the generating relations of the (−1)-shifted derived Poisson algebra structure on Ω • F (∧ • B), are exactly those of[2, Proposition 4.3]. This concludes the proof.…”

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confidence: 67%

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Atiyah and Todd classes arising from integrable distributions

Chen

Xiang

2019

Journal of Geometry and Physics

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supporting

confidence: 67%

“…is a differential Gerstenhaber algebra, thus also a (−1)-shifted derived Poisson algebra [2], where [−, −] SN is the Schouten-Nijenhuis bracket. Op cit, a (−1)-shifted derived Poisson algebra structure is constructed on Γ(M, ∧ • F ∨ ⊗ ∧ • (T K M/F )).…”

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confidence: 99%

Atiyah and Todd classes arising from integrable distributions

Chen

Xiang

2019

Journal of Geometry and Physics

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“…Note that both (C • reg (E), d E ) and (C • (A E ), d CE ) are commutative dg algebras. It is natural to expect that this contraction is compatible with the relevant algebra structures, in other words, a semifull algebra contraction introduced by Real [40] (see also [4] for an equivalent definition). We will return to this problem in the future.…”

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confidence: 99%

The standard cohomology of regular Courant algebroids

Cai¹,

Chen²,

Xiang³

2021

Preprint

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For any Courant algebroid E over a smooth manifold M with characteristic distribution F which is regular, we study the standard cohomology H • st (E) by using a special spectral sequence. We prove a theorem which tells how a natural transgression map [T ] together with the Chevalley-Eilenberg cohomology of the ample Lie algebroid A E of E with coefficient in the symmetric tensor product S(T M/F ) of the normal bundle T M/F determines H • st (E), thereby significantly reducing the range of generators of the standard cohomology. We can also recover an earlier result by Ginot and Grützmann (J. Symplectic Geom. 7 (2009), no. 3, 311-335) in the situation that the base manifold M splits. In addition, we apply the theorem to two special types of regular Courant algebroids: generalized exact ones and those arising from regular Lie algebroids.

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“…To prove the second statement of Theorem 4.4, we need the homological perturbation lemma (cf. [4]), which we recall as follows:…”

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confidence: 99%

Cohomology of hemistrict Lie 2-algebras

Cai

Liu

Xiang

2020

Communications in Algebra

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We study representations of hemistrict Lie 2-algebras and give a functorial construction of their cohomology. We prove that both the cohomology of an injective hemistrict Lie 2-algebra L and the cohomology of the semistrict Lie 2-algebra obtained from skew-symmetrization of L are isomorphic to the Chevalley-Eilenberg cohomology of the induced Lie algebra L Lie .[−, −] g induces a semistrict Lie 2-algebra G = (K[1] → g, l 2 , l 3 ). As a consequence, we haveTheorem (see Theorem 4.12). Let g be a Leibniz algebra with Leibniz kernel K and V a representation of the associated hemistrict Lie 2-algebra L g to g such that l α = 0 for all α ∈ K. ThenThe sequel(s). We plan to write two sequels to this paper, in which we address several issues not covered here: In the first one in preparation, we study cohomology of weak Lie 2-algebras, which encodes cohomology of hemistrict Lie 2-algebras and semistrict Lie 2-algebras into a unified framework. Our first goal is to establish the compatibility between the functor of taking cohomology and the skew-symmetrization functor [18] from the category of weak Lie 2-algebras to the category of semistrict Lie 2-algebras. Meanwhile, in their work [10] of weak Lie 2-bialgebras, Chen, Stiénon and Xu developed an odd version of big derived bracket approach to semistrict Lie 2-algebras. Our second goal is to establish a derived bracket formalism for cohomology of weak Lie 2-algebras.In the second one, we consider weak L ∞ -algebroids which encodes a Courant algebroid as a 2-term weak L ∞ -algebroid. According to Kontsevich and Soibelman [12], L ∞ -algebras correspond to formal pointed dg manifolds, while A ∞ -algebras correspond to noncommutative formal pointed dg manifolds. Our purposes are to reinterpret weak L ∞ -algebroids as certain commutative up to homotopy formal dg manifolds and to investigate their relation with shifted derived Poisson manifolds studied in [4] and strongly homotopy Leibniz algebra over a commutative dg algebra studied in [9]. Representations of hemistrict Lie 2-algebrasIn this section, we study representations of hemistrict Lie 2-algebras on 2-term cochain complexes as Beck modules of the category of hemistrict Lie 2-algebras.2.1. Dg Leibniz algebras and hemistrict Lie 2-algebras. As the first step, we collect some basic facts of dg Leibniz algebras, which are Leibniz algebras (or Loday algebras) introduced by Loday [16] in the category of cochain complexes (see [11] for more structure theorems on Leibniz algebras). Denote by K the base field which is either R or C.Definition 2.1. A dg (left) Leibniz algebra g over K is a (left) Leibniz algebra in the category of cochain complexes of K-vector spaces, i.e., a cochain complex g = (g • , d) together with a cochain map [−, −] g : g ⊗ g → g, called Leibniz bracket, satisfying the (left) Leibniz rule:Example 2.2. Let h = (h • , d, [−, −] h ) be a dg Lie algebra equipped with a representation ρ on a cochain complex V = (V • , d). Then the direct sum cochain complex h ⊕ V = (h • ⊕ V • , d), equipped with the bilinear map ...

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Shifted Derived Poisson Manifolds Associated with Lie Pairs

Cited by 19 publications

References 48 publications

Atiyah and Todd classes arising from integrable distributions

Atiyah and Todd classes arising from integrable distributions

The standard cohomology of regular Courant algebroids

Cohomology of hemistrict Lie 2-algebras

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