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 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.

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.

2-PLECTIC MANIFOLDS AND ITS RELATION WITH COURANT ALGEBROIDS

2012 ◽  
Vol 09 (03) ◽  
pp. 1250015
Author(s):  
M. SHAFIEE
Keyword(s):  
Lie Group ◽  
Lagrangian Submanifolds ◽  
Compact Lie Group ◽  
Courant Algebroids ◽  
Courant Algebroid ◽  
Hamiltonian Group

In this paper we study the relation between 2-plectic manifolds and Courant algebroids. We establish a relation between 2-Lagrangian submanifolds of 2-plectic manifolds and subbundles of Courant algebroids. Also we show that an action of a compact Lie group G on a 2-plectic manifold (M, ω) can be extended to an action of G on an exact Courant algebroid E over M if and only if G is a subgroup of Hamiltonian group of (M, ω).

scholarly journals Matched pairs of Courant algebroids

2014 ◽  
Vol 25 (5) ◽  
pp. 977-991
Author(s):  
Melchior Grützmann ◽  
Mathieu Stiénon
Keyword(s):  
Matched Pairs ◽  
Courant Algebroids

scholarly journals CONFORMAL COURANT ALGEBROIDS AND ORIENTIFOLD T-DUALITY

2012 ◽  
Vol 10 (02) ◽  
pp. 1250084 ◽  
Author(s):  
DAVID BARAGLIA
Keyword(s):  
Line Bundle ◽  
Bilinear Form ◽  
Conformal Structure ◽  
Crossed Module ◽  
Courant Algebroids ◽  
Courant Algebroid ◽  
Circle Bundles ◽  
Twisted Cohomology ◽  
Free Involution ◽  
Special Case

We introduce conformal Courant algebroids, a mild generalization of Courant algebroids in which only a conformal structure rather than a bilinear form is assumed. We introduce exact conformal Courant algebroids and show they are classified by pairs (L, H) with L a flat line bundle and H ∈ H3(M, L) a degree 3 class with coefficients in L. As a special case gerbes for the crossed module (U(1) → ℤ2) can be used to twist TM ⊕ T*M into a conformal Courant algebroid. In the exact case there is a twisted cohomology which is 4-periodic if L2 = 1. The structure of Conformal Courant algebroids on circle bundles leads us to construct a T-duality for orientifolds with free involution. This incarnation of T-duality yields an isomorphism of 4-periodic twisted cohomology. We conjecture that the isomorphism extends to an isomorphism in twisted KR-theory and give some calculations to support this claim.

scholarly journals Transitive double Lie algebroids via core diagrams

2021 ◽  
Vol 13 (3) ◽  
pp. 403
Author(s):  
Madeleine Jotz Lean ◽  
Kirill C. H. Mackenzie
Keyword(s):  
Lie Algebroids ◽  
Lie Algebroid ◽  
Lie Groupoids ◽  
Base Manifold ◽  
Lie Groupoid ◽  
The Core ◽  
Fixed Base ◽  
The One

<p style='text-indent:20px;'>The core diagram of a double Lie algebroid consists of the core of the double Lie algebroid, together with the two core-anchor maps to the sides of the double Lie algebroid. If these two core-anchors are surjective, then the double Lie algebroid and its core diagram are called <i>transitive</i>. This paper establishes an equivalence between transitive double Lie algebroids, and transitive core diagrams over a fixed base manifold. In other words, it proves that a transitive double Lie algebroid is completely determined by its core diagram.</p><p style='text-indent:20px;'>The comma double Lie algebroid associated to a morphism of Lie algebroids is defined. If the latter morphism is one of the core-anchors of a transitive core diagram, then the comma double algebroid can be quotiented out by the second core-anchor, yielding a transitive double Lie algebroid, which is the one that is equivalent to the transitive core diagram.</p><p style='text-indent:20px;'>Brown's and Mackenzie's equivalence of transitive core diagrams (of Lie groupoids) with transitive double Lie groupoids is then used in order to show that a transitive double Lie algebroid with integrable sides and core is automatically integrable to a transitive double Lie groupoid.</p>

scholarly journals Issues for the Global Construction Market

2012 ◽  
Vol 1 (2) ◽  
pp. 73-81 ◽  
Author(s):  
George Najjir ◽  
Philip Love ◽  
Goran Runeson
Keyword(s):  
Conceptual Framework ◽  
Construction Industry ◽  
Ownership Structure ◽  
Current Understanding ◽  
Industry Development ◽  
Construction Market ◽  
The Core ◽  
Classic Theory ◽  
New Construction ◽  
Global Construction

This paper examines issues of the global construction environment. The core theses developed is that globalization will create a new construction industry. A conceptual framework initiates discussion in relation to four selected perspectives vis. neo-classic theory; a comparison with manufacturing; construction industry development; and ownership structure. Conclusions drawn indicate that analysis is only just emerging, and that future examination will require a shift in current understanding so as to consider the global firm as operating in a new and totally separate market.

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

2018 ◽  
Vol 61 (3) ◽  
pp. 588-607 ◽  
Author(s):  
Honglei Lang ◽  
Yunhe Sheng ◽  
Aïssa Wade
Keyword(s):  
Lie Algebra ◽  
Complex Structure ◽  
Contact Structures ◽  
Complex Structures ◽  
Courant Algebroids ◽  
Courant Algebroid ◽  
Generalized Complex Structures ◽  
Generalized Complex Structure ◽  
Generalized Geometry ◽  
Generalized Complex

AbstractIn this paper, we first discuss the relation between VB-Courant algebroids and E-Courant algebroids, and we construct some examples of E-Courant algebroids. Then 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.

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
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