Hom-Lie algebroids

Laurent-Gengoux, Camille; Teles, Joana

doi:10.1016/j.geomphys.2013.02.003

Cited by 37 publications

(53 citation statements)

References 9 publications

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“…In this section, we recall some basic definitions from [8], [11], and [13]. Let R be a commutative ring with unity and Z + be the set of all non-negative integers.…”

Section: Preliminaries On Hom-structuresmentioning

confidence: 99%

“…More recently, the notion of hom-Lie algebroid is introduced in [8] by going through a formulation of hom-Gerstenhaber algebra and following the classical bijective correspondence between Lie algebroids and Gerstenhaber algebras. On the other hand, there are canonically defined adjoint functors between category of Lie-Rinehart algebras and category of Gerstenhaber algebras.…”

Section: Introductionmentioning

confidence: 99%

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On Hom-Gerstenhaber algebras, and Hom-Lie algebroids

Mandal

Mishra

2018

Journal of Geometry and Physics

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We define the notion of hom-Batalin-Vilkovisky algebras and strong differential hom-Gerstenhaber algebras as a special class of hom-Gerstenhaber algebras and provide canonical examples associated to some well-known hom-structures. Representations of a hom-Lie algebroid on a hom-bundle are defined and a cohomology of a regular hom-Lie algebroid with coefficients in a representation is studied. We discuss about relationship between these classes of hom-Gerstenhaber algebras and geometric structures on a vector bundle. As an application, we associate a homology to a regular hom-Lie algebroid and then define a hom-Poisson homology associated to a hom-Poisson manifold.

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“…In this section, we recall some basic definitions from [8], [11], and [13]. Let R be a commutative ring with unity and Z + be the set of all non-negative integers.…”

Section: Preliminaries On Hom-structuresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Hom-Gerstenhaber algebras, and Hom-Lie algebroids

Mandal

Mishra

2018

Journal of Geometry and Physics

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“…Now, we introduce two kinds of definitions of Hom-Lie algebroids, they are from [7] and [13]. More about Hom-Lie algebroids, please see [7] and [13].…”

Section: Hom-lie Algebroidsmentioning

confidence: 99%

“…In [15], there is a series of coboundary operators, and the author generalizes the result " If k is a Lie algebra, ρ : k −→ gl(V ) is a representation if and only if there is a degree-1 operator D on Λk * ⊗ V satisfying D 2 = 0, andwhere d k : ∧ k g * −→ ∧ k+1 g * is the coboundary operator associated to the trivial representation. "Geometric generalizations of Hom-Lie algebras are given in [7][13]. In [7], C. Laurent-Gengoux and J. Teles proved that there is a one-to-one correspondence between Hom-Gerstenhaber algebras and Hom-Lie algebroids; in [14], base on Hom-Lie algebroids from [7], the authors study representation of Hom-Lie algebroids.…”

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confidence: 99%

Equivalent Description of Hom-Lie Algebroids

Xiong

2018

Advances in Mathematical Physics

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In this paper, we study representations of Hom-Lie algebroids, give some properties of Hom-Lie algebroids and discuss equivalent statements of Hom-Lie algebroids. Then, we prove that two known definitions of Hom-Lie algebroids can be transformed into each other under some conditions. IntroductionThe notion of Hom-Lie algebras was introduced by Hartwig, Larsson, and Silvestrov in [3] as a part of a study of deformations of the Witt and the Virasoro algebras. In a Hom-Lie algebra, the Jacobi identity is twisted by a linear map, called the Hom-Jacobi identity. Some q-deformations of the Witt and the Virasoro algebras have the structure of a Hom-Lie algebra [3,4]. Because of close relations to discrete and deformed vector fields and differential calculus [3, 5, 6], more people pay special attention to this algebraic structure. For a party of k-cochains on Hom-Lie algebras, name k-Hom-cochains, there is a series of coboundary operators [11]; for regular Hom-Lie algebras, [12] gives a new coboundary operator on k-cochains, and there are many works have been done by the special coboundary operator [12,13]. In [15], there is a series of coboundary operators, and the author generalizes the result " If k is a Lie algebra, ρ : k −→ gl(V ) is a representation if and only if there is a degree-1 operator D on Λk * ⊗ V satisfying D 2 = 0, andwhere d k : ∧ k g * −→ ∧ k+1 g * is the coboundary operator associated to the trivial representation."Geometric generalizations of Hom-Lie algebras are given in [7][13]. In [7], C. Laurent-Gengoux and J. Teles proved that there is a one-to-one correspondence between Hom-Gerstenhaber algebras and Hom-Lie algebroids; in [14], base on Hom-Lie algebroids from [7], the authors study representation of Hom-Lie algebroids. In [13], the authors make small modifications to the definition of Hom-Lie algebroids, and give a new definition of Hom-Lie algebroids, base on the new definition of Hom-Lie algebroids, definitions of Hom-Lie bialgebroids and Hom-Courant algebroids are given.In this article, we first study representations of Hom-Lie algebroids, give equivalent statements of Hom-Lie algebroids and prove that different definitions of Hom-Lie algebroids are given by the same Hom-Lie algebras and their representations. 0 Keyword: Hom-Lie algebroids; Hom-Lie algebras; representations 0 MSC: 17B99,58H05 0

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“…In [14], hom-Lie-Rinehart algebras are introduced as an algebraic analogue of hom-Lie algebroids. This notion generalises both the notion of a hom-Lie algebra and the notion of a hom-Lie algebroid in [9]. By starting with a Lie-Rinehart algebra ( [15], [5], [6], [7]) and a homomorphism into itself, one obtains a canonically associated hom-Lie-Rinehart algebra (usually referred as "obtained by composition").…”

Section: Introductionmentioning

confidence: 99%