scholarly journals Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids

2017 ◽  
Vol 121 ◽  
pp. 15-32 ◽  
Author(s):  
Liqiang Cai ◽  
Jiefeng Liu ◽  
Yunhe Sheng
Keyword(s):  
Lie Algebroids ◽  
Courant Algebroids

scholarly journals On LA-Courant Algebroids and Poisson Lie 2-Algebroids

2020 ◽  
Vol 23 (3) ◽  
Author(s):  
M. Jotz Lean
Keyword(s):  
Lie Algebroids ◽  
General Context ◽  
Matched Pairs ◽  
Courant Algebroids ◽  
Dirac Structures ◽  
Courant Algebroid ◽  
The Core ◽  
Precise Sense ◽  
Nondegenerate Case ◽  
New Construction

Abstract This paper reformulates Li-Bland’s definition for LA-Courant algebroids, or Poisson Lie 2-algebroids, in terms of split Lie 2-algebroids and self-dual 2-representations. This definition generalises in a precise sense the characterisation of (decomposed) double Lie algebroids via matched pairs of 2-representations. We use the known geometric examples of LA-Courant algebroids in order to provide new examples of Poisson Lie 2-algebroids, and we explain in this general context Roytenberg’s equivalence of Courant algebroids with symplectic Lie 2-algebroids. We study further the core of an LA-Courant algebroid and we prove that it carries an induced degenerate Courant algebroid structure. In the nondegenerate case, this gives a new construction of a Courant algebroid from the corresponding symplectic Lie 2-algebroid. Finally we completely characterise VB-Dirac and LA-Dirac structures via simpler objects, that we compare to Li-Bland’s pseudo-Dirac structures.

scholarly journals Higher extensions of Lie algebroids

2017 ◽  
Vol 19 (03) ◽  
pp. 1650034 ◽  
Author(s):  
Yunhe Sheng ◽  
Chenchang Zhu
Keyword(s):  
Lie Algebroids ◽  
Lie Algebroid ◽  
Courant Algebroids ◽  
Courant Algebroid ◽  
Semidirect Products

We study the extension of a Lie algebroid by a representation up to homotopy, including semidirect products of a Lie algebroid with such representations. The extension results in a higher Lie algebroid. We give exact Courant algebroids and string Lie 2-algebras as examples of such extensions. We then apply this to obtain a Lie 2-groupoid integrating an exact Courant algebroid.

scholarly journals Modular classes revisited

2014 ◽  
Vol 11 (09) ◽  
pp. 1460042 ◽  
Author(s):  
Janusz Grabowski
Keyword(s):  
Geometric Approach ◽  
Geometric Interpretation ◽  
Lie Algebroids ◽  
Poisson Geometry ◽  
Dirac Structure ◽  
Poisson Structures ◽  
Courant Algebroids ◽  
Modular Class ◽  
Novel Approach

We present a graded-geometric approach to modular classes of Lie algebroids and their generalizations, introducing in this setting an idea of relative modular class of a Dirac structure for certain type of Courant algebroids, called projectable. This novel approach puts several concepts related to Poisson geometry and its generalizations in a new light and simplifies proofs. It gives, in particular, a nice geometric interpretation of modular classes of twisted Poisson structures on Lie algebroids.

L∞-actions of Lie algebroids

2021 ◽  
pp. 2150013
Author(s):  
Olivier Brahic ◽  
Marco Zambon
Keyword(s):  
Vector Field ◽  
Direct Product ◽  
Vector Fields ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Geometric Structures ◽  
Courant Algebroids ◽  
Graded Manifold

We consider homotopy actions of a Lie algebroid on a graded manifold, defined as suitable [Formula: see text]-algebra morphisms. On the “semi-direct product” we construct a homological vector field that projects to the Lie algebroid. Our main theorem states that this construction is a bijection. Since several classical geometric structures can be described by homological vector fields as above, we can display many explicit examples, involving Lie algebroids (including extensions, representations up to homotopy and their cocycles) as well as transitive Courant algebroids.

Banach Lie Algebroids

2011 ◽  
Vol 57 (2) ◽  
pp. 409-416
Author(s):  
Mihai Anastasiei
Keyword(s):  
Differential Equations ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Second Order ◽  
Lie Algebroids ◽  
Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

Higher nonabelian omni-Lie algebroids

2021 ◽  
pp. 104277
Author(s):  
Bi Yanhui ◽  
Fan Hongtao
Keyword(s):  
Lie Algebroids

scholarly journals Exotic Courant algebroids and T-duality

2021 ◽  
Vol 163 ◽  
pp. 104155
Author(s):  
Jaklyn Crilly ◽  
Varghese Mathai
Keyword(s):  
Courant Algebroids

scholarly journals Gauged sigma-models with nonclosed 3-form and twisted Jacobi structures

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Athanasios Chatzistavrakidis ◽  
Grgur Šimunić
Keyword(s):  
Sigma Models ◽  
Sigma Model ◽  
Target Space ◽  
Two Dimensional ◽  
Courant Algebroids ◽  
Dirac Structures ◽  
Courant Algebroid ◽  
Abelian Case ◽  
Jacobi Manifolds ◽  
Jacobi Structure

Abstract We study aspects of two-dimensional nonlinear sigma models with Wess-Zumino term corresponding to a nonclosed 3-form, which may arise upon dimensional reduction in the target space. Our goal in this paper is twofold. In a first part, we investigate the conditions for consistent gauging of sigma models in the presence of a nonclosed 3-form. In the Abelian case, we find that the target of the gauged theory has the structure of a contact Courant algebroid, twisted by a 3-form and two 2-forms. Gauge invariance constrains the theory to (small) Dirac structures of the contact Courant algebroid. In the non-Abelian case, we draw a similar parallel between the gauged sigma model and certain transitive Courant algebroids and their corresponding Dirac structures. In the second part of the paper, we study two-dimensional sigma models related to Jacobi structures. The latter generalise Poisson and contact geometry in the presence of an additional vector field. We demonstrate that one can construct a sigma model whose gauge symmetry is controlled by a Jacobi structure, and moreover we twist the model by a 3-form. This construction is then the analogue of WZW-Poisson structures for Jacobi manifolds.

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