scholarly journals On Lie bialgebroid crossed modules

2021 ◽  
Vol 32 (04) ◽  
pp. 2150021
Author(s):  
Honglei Lang ◽  
Yu Qiao ◽  
Yanbin Yin
Keyword(s):  
Lie Algebroid ◽  
Dirac Structures ◽  
Crossed Modules ◽  
Manin Triples

We study Lie bialgebroid crossed modules which are pairs of Lie algebroid crossed modules in duality that canonically give rise to Lie bialgebroids. A one-one correspondence between such Lie bialgebroid crossed modules and co-quadratic Manin triples [Formula: see text] is established, where [Formula: see text] is a co-quadratic Lie algebroid and [Formula: see text] is a pair of transverse Dirac structures in [Formula: see text].

scholarly journals Dirac Structures on Banach Lie Algebroids

2014 ◽  
Vol 22 (3) ◽  
pp. 219-228
Author(s):  
Vlad-Augustin Vulcu
Keyword(s):  
Vector Bundle ◽  
Tangent Bundle ◽  
Differential Calculus ◽  
Vector Bundles ◽  
Dirac Structure ◽  
Lie Algebroid ◽  
Symmetric Form ◽  
Courant Bracket ◽  
Dirac Structures ◽  
Original Definition

Abstract In the original definition due to A. Weinstein and T. Courant a Dirac structure is a subbundle of the big tangent bundle T M ⊕ T* M that is equal to its ortho-complement with respect to the so-called neutral metric on the big tangent bundle. In this paper, instead of the big tangent bundle we consider the vector bundle E ⊕ E*, where E is a Banach Lie algebroid and E* its dual. Recall that E* is not in general a Lie algebroid. We define a bilinear and symmetric form on the vector bundle E ⊕ E* and say that a subbundle of it is a Dirac structure if it is equal with its orthocomplement. Our main result is that any Dirac structure that is closed with respect to a type of Courant bracket, endowed with a natural anchor is a Lie algebroid. In the proof the differential calculus on a Lie algebroid is essentially involved. We work in the category of Banach vector bundles.

scholarly journals Lie algebroid foliations and $\Script E$1(M)-Dirac structures

2002 ◽  
Vol 35 (18) ◽  
pp. 4085-4104 ◽  
Author(s):  
David Iglesias ◽  
Juan C Marrero
Keyword(s):  
Lie Algebroid ◽  
Dirac Structures

scholarly journals DIRAC STRUCTURES OF OMNI-LIE ALGEBROIDS

2011 ◽  
Vol 22 (08) ◽  
pp. 1163-1185 ◽  
Author(s):  
ZHUO CHEN ◽  
ZHANG JU LIU ◽  
YUNHE SHENG
Keyword(s):  
Lie Algebras ◽  
Cohomology Group ◽  
Adjoint Representation ◽  
Lie Algebroids ◽  
Reduction Theory ◽  
Dirac Structure ◽  
Lie Algebroid ◽  
Dirac Structures ◽  
Derivation Algebra ◽  
One To One

Omni-Lie algebroids are generalizations of Alan Weinstein's omni-Lie algebras. A Dirac structure in an omni-Lie algebroid 𝔇E ⊕ 𝔍E is necessarily a Lie algebroid together with a representation on E. We study the geometry underlying these Dirac structures in the light of reduction theory. In particular, we prove that there is a one-to-one correspondence between reducible Dirac structures and projective Lie algebroids in [Formula: see text]; we establish the relation between the normalizer NL of a reducible Dirac structure L and the derivation algebra Der (b (L)) of the projective Lie algebroid b(L); we study the cohomology group H •(L, ρL) and the relation between NL and H 1(L, ρL); we describe Lie bialgebroids using the adjoint representation; we study the deformation of a Dirac structure L, which is related with H 2(L, ρL).

scholarly journals Abelian extensions and crossed modules of Hom-Lie algebras

2020 ◽  
Vol 224 (3) ◽  
pp. 987-1008
Author(s):  
José Manuel Casas ◽  
Xabier García-Martínez
Keyword(s):  
Lie Algebras ◽  
Abelian Extensions ◽  
Crossed Modules

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.

Crossed Semimodules and Schreier Internal Categories in the Category of Monoids

1998 ◽  
Vol 5 (6) ◽  
pp. 575-581 ◽  
Author(s):  
A. Patchkoria
Keyword(s):  
Internal Category ◽  
Equivalence Of Categories ◽  
Crossed Modules

Abstract We introduce the notion of a Schreier internal category in the category of monoids and prove that the category of Schreier internal categories in the category of monoids is equivalent to the category of crossed semimodules. This extends a well-known equivalence of categories between the category of internal categories in the category of groups and the category of crossed modules.

scholarly journals Computation and homotopical applications of induced crossed modules

2003 ◽  
Vol 35 (1) ◽  
pp. 59-72 ◽  
Author(s):  
Ronald Brown ◽  
Christopher D. Wensley
Keyword(s):  
Crossed Modules
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