Omni n-Lie algebras and linearization of higher analogues of Courant algebroids

Liu, Jiefeng; Sheng, Yunhe; Wang, Chunyue

doi:10.1142/s0219887817501134

Cited by 3 publications

(7 citation statements)

References 26 publications

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“…• When the vector bundle reduces to a vector space, it reduces to the omni n-Lie algebra introduced in [26];…”

Section: Resultsmentioning

confidence: 99%

“…Recently, the higher analogues of the standard Courant algebroid T M ⊕ ∧ n T * M are widely studied due to applications in Nambu-Poisson structures, multisymplectic structures, L ∞ -algebra theory and topological field theory [1,3,12,14,15,17,37]. In [26], the authors introduced the notion of an omni n-Lie algebra gl(V ) ⊕ ∧ n V and proved that it is the base-linearization of the higher analogue of the standard Courant algebroid T M ⊕ ∧ n T * M . Moreover, the (n + 1)-Lie algebra structures on V can be characterized as integrable subspaces of the omni n-Lie algebra gl(V ) ⊕ ∧ n V .…”

Section: Omni N-lie Algebras and N-omni-lie Algebroidsmentioning

confidence: 99%

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Linearization of the higher analogue of Courant algebroids

Lang,

Sheng

2020

Preprint

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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 J n E 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 ⊕ J n E introduced by Bi-Vitagliago-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. Contents 1 Introduction

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“…• When the vector bundle reduces to a vector space, it reduces to the omni n-Lie algebra introduced in [26];…”

Section: Resultsmentioning

confidence: 99%

Section: Omni N-lie Algebras and N-omni-lie Algebroidsmentioning

confidence: 99%

Linearization of the higher analogue of Courant algebroids

Lang,

Sheng

2020

Preprint

Self Cite

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“…In particular, Dirac structures of the higher analogues of the standard Courant algebroid T M ⊕ ∧ n T * M are deeply studied in [2,6,20,42]. In [31], the authors introduced the notion of an omni n-Lie algebra gl(V ) ⊕ ∧ n V and proved that it is the base-linearization of the higher analogue of the standard Courant algebroid T M ⊕ ∧ n T * M . Moreover, the (n + 1)-Lie algebra structures on V can be characterized as integrable subspaces of the omni n-Lie algebra gl(V ) ⊕ ∧ n V .…”

Section: 2mentioning

confidence: 99%

“…When E = V , a vector space, the n-omni-Lie algebroid for n ≥ 2 is just gl(V ). So n-omni-Lie algebroids do not include omni n-Lie algebras studied in [31] as special cases.…”

Section: 2mentioning

confidence: 99%

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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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Higher nonabelian omni-Lie algebroids

Fan

2021

Journal of Geometry and Physics

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Omni n-Lie algebras and linearization of higher analogues of Courant algebroids

Cited by 3 publications

References 26 publications

Linearization of the higher analogue of Courant algebroids

Linearization of the higher analogue of Courant algebroids

Linearization of the higher analogue of Courant algebroids

Higher nonabelian omni-Lie algebroids

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