A Hopf algebra associated with a Lie pair

Chen, Zhuo; Stiénon, Mathieu; Xu, Ping

doi:10.1016/j.crma.2014.09.010

Cited by 7 publications

(8 citation statements)

References 9 publications

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“…It is indeed a quasi-isomorphism of complexes of F -modules [12,24]. Therefore, it induces an isomorphism of vector spaces hkr :…”

Section: 13mentioning

confidence: 99%

Hochschild cohomology of dg manifolds associated to integrable distributions

Chen,

Xiang,

2021

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For the field K = R or C, and an integrable distribution F ⊆ T K M = TM ⊗ R K on a smooth manifold M , we study the Hochschild cohomology of the dg manifold (F [1], dF ) and establish a canonical isomorphism with the Hochschild cohomology of the transversal polydifferential operators of F . In particular, for the dg manifold (T 0,1 X [1], ∂) associated with a complex manifold X, we prove that it is canonically isomorphic to the Hochschild cohomology HH • (X) of the complex manifold. As an application, we show that the Kontsevich-Duflo type theorem for the dg manifold (T 0,1 X [1], ∂) implies the Kontsevich-Duflo theorem for complex manifolds.

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“…It is indeed a quasi-isomorphism of complexes of F -modules [12,24]. Therefore, it induces an isomorphism of vector spaces hkr :…”

Section: 13mentioning

confidence: 99%

Hochschild cohomology of dg manifolds associated to integrable distributions

Chen,

Xiang,

2021

Preprint

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“…We are now ready to turn to We remark that similar results appeared in [12] for that of Lie pairs, and in [7] of relative Lie algebroids. In the meantime, the work related to some facts in the derived categories claimed in [11] is still going on. Let the dual vectors be a ∨ i , c ∨ and b ∨ , with degrees…”

Section: Atiyah Classes As Functorsmentioning

confidence: 99%

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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“…In the last decade much research on Lie pairs has been done following different strategies and the underlying mathematical structures: Atiyah classes arising from Lie pairs have been studied, using a variety of methods, see e.g. [2,8,9]; It is shown that geometric objects including Kapranov dg and Fedosov dg manifolds [19,30], algebraic objects such as Hopf algebras [7,11], Leibniz ∞ and L ∞ algebras can be derived from Lie pairs [1,6,20]; Also, in the context of Lie pairs, considerable attentions had been paid to Poincaré-Birkhoff-Witt isomorphisms [4,5], Kontsevich-Duflo isomorphisms [10,21], and Rozansky-Witten-type invariants [37], etc.…”

Section: Introductionmentioning

confidence: 99%

Internal symmetry of the $L_{\leqslant 3}$ algebra arising from a Lie pair

Ni¹,

Cheng²,

Chen³

et al. 2022

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A Lie pair is an inclusion A to L of Lie algebroids over the same base manifold. In an earlier work, the third author with Bandiera, Stiénon, and Xu introduced a canonical L 3 algebra Γ(∧ • A ∨ ) ⊗ Γ(L/A) whose unary bracket is the Chevalley-Eilenberg differential arising from every Lie pair (L, A). In this note, we prove that to such a Lie pair there is an associated Lie algebra action by Diff(L) on the L 3 algebra Γ(∧ • A ∨ ) ⊗ Γ(L/A). Here Diff(L) is the space of 1-differentials on the Lie algebroid L, or infinitesimal automorphisms of L. The said action gives rise to a larger scope of gauge equivalences of Maurer-Cartan elements in Γ(∧ • A ∨ ) ⊗ Γ(L/A), and for this reason we elect to call the Diff(L)-action internal symmetry of Γ(∧ • A ∨ ) ⊗ Γ(L/A).

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A Hopf algebra associated with a Lie pair

Cited by 7 publications

References 9 publications

Hochschild cohomology of dg manifolds associated to integrable distributions

Hochschild cohomology of dg manifolds associated to integrable distributions

Atiyah Classes of Strongly Homotopy Lie Pairs

Internal symmetry of the $L_{\leqslant 3}$ algebra arising from a Lie pair

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