On Deformations of Lie Algebroids

Sheng, Yunhe

doi:10.1007/s00025-011-0133-x

Cited by 15 publications

(10 citation statements)

References 21 publications

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“…We shall generalize this result to linear n-vector fields. Actually, linear n-vector fields studied in [5,25] are isomorphic to sections of the n-th differential operator bundle D n E introduced in [10,35].…”

Section: Resultsmentioning

confidence: 99%

Linearization of the higher analogue of Courant algebroids

Lang¹,

Sheng²

2020

Journal of Geometric Mechanics

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In this paper, we show that the spaces of sections of the n-th differential operator bundle D n E and the n-th skew-symmetric jet bundle JnE of a vector bundle E are isomorphic to the spaces of linear n-vector fields and linear n-forms on E * respectively. Consequently, the n-omni-Lie algebroid DE ⊕ JnE introduced by Bi-Vitagliano-Zhang can be explained as certain linearization, which we call pseudo-linearization of the higher analogue of Courant algebroids T E * ⊕ ∧ n T * E *. On the other hand, we show that the omni n-Lie algebroid DE ⊕ ∧ n JE can also be explained as certain linearization, which we call Weinstein-linearization of the higher analogue of Courant algebroids T E * ⊕ ∧ n T * E *. We also show that n-Lie algebroids, local n-Lie algebras and Nambu-Jacobi structures can be characterized as integrable subbundles of omni n-Lie algebroids.

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Section: Resultsmentioning

confidence: 99%

Linearization of the higher analogue of Courant algebroids

Lang¹,

Sheng²

2020

Journal of Geometric Mechanics

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“…Vector bundles Hom(∧ k DE, E) JE and Hom(∧ k JE, E) DE are introduced in [5] and [21] to study deformations of omni-Lie algebroids and deformations of Lie algebroids respectively. More precisely, we have…”

Section: Generalized Complex Structures On Omni-lie Algebroidsmentioning

confidence: 99%

VB-Courant Algebroids, E-Courant Algebroids and Generalized Geometry

Lang

Sheng

Wade³

2018

Can. math. bull.

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Abstract. In this paper, we rst discuss the relation between VB-Courant algebroids and E-Courant algebroids and construct some examples of E-Courant algebroids.en we introduce the notion of a generalized complex structure on an E-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra gl(V ) ⊕ V correspond to complex Lie algebra structures on V .

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“…In this section, we introduce another cochain complex associated to a left‐symmetric algebroid, which could control deformations. See , for deformations of Lie algebroids. Definition Let E be a vector bundle over M , a multiderivation of degree n is a multilinear map

D \in Hom (Λ^{n - 1} Γ (E) \otimes Γ (E), Γ (E))

, such that for all

f \in C^{\infty} (M)

and sections

x_{i} \in Γ (E)

, the following conditions are satisfied:

\begin{matrix} true \\ right D (x_{1}, ..., f x_{i}, ..., x_{n - 1}, x_{n}) & = & left f D (x_{1}, ..., x_{i}, ..., x_{n - 1}, x_{n}), i = 1, ..., n - 1; \\ right D (x_{1}, ..., x_{n - 1}, f x_{n}) & = & left f D (x_{1}, ..., x_{n - 1}, x_{n}) + σ_{D} (x_{1}, ..., x_{n - 1}) (f) x_{n}, \end{matrix}

where

σ_{D} \in Γ (Hom (Λ^{n - 1} E, T M))

is called the symbol .…”

Section: Deformation Cohomologies Of Left‐symmetric Algebroidsmentioning

confidence: 99%

“…In this section, we introduce another cochain complex associated to a left-symmetric algebroid, which could control deformations. See [9], [20] for deformations of Lie algebroids. Definition 6.1 Let E be a vector bundle over M, a multiderivation of degree n is a multilinear map D ∈ Hom n−1 (E) ⊗ (E), (E) , such that for all f ∈ C ∞ (M) and sections x i ∈ (E), the following conditions are satisfied: D(x 1 , .…”

Section: Deformation Cohomologies Of Left-symmetric Algebroidsmentioning

confidence: 99%

Left‐symmetric algebroids

Liu

Sheng

Bai

et al. 2016

Mathematische Nachrichten

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In this paper, we introduce a notion of a left-symmetric algebroid, which is a generalization of a left-symmetric algebra from a vector space to a vector bundle. The left multiplication gives rise to a representation of the corresponding sub-adjacent Lie algebroid. We construct left-symmetric algebroids from O-operators on Lie algebroids. We study phase spaces of Lie algebroids in terms of left-symmetric algebroids. Representations of left-symmetric algebroids are studied in detail. At last, we study deformations of left-symmetric algebroids, which could be controlled by the second cohomology class in the deformation cohomology.

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On Deformations of Lie Algebroids

Cited by 15 publications

References 21 publications

Linearization of the higher analogue of Courant algebroids

Linearization of the higher analogue of Courant algebroids

VB-Courant Algebroids, E-Courant Algebroids and Generalized Geometry

Left‐symmetric algebroids

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