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

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].

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Dirac Structures on Banach Lie Algebroids

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2014-0060 ◽

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.

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DIRAC STRUCTURES OF OMNI-LIE ALGEBROIDS

International Journal of Mathematics ◽

10.1142/s0129167x11007215 ◽

2011 ◽

Vol 22 (08) ◽

pp. 1163-1185 ◽

Cited By ~ 5

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).

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

Journal of High Energy Physics ◽

10.1007/jhep11(2020)173 ◽

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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