<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In the original definition due to A. Weinstein and T. Courant a Dirac structure is a subbundle of the big tangent bundle <italic>T M</italic> ⊕ <italic>T* M</italic> 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 <italic>E</italic> ⊕ <italic>E*</italic>, where E is a Banach Lie algebroid and <italic>E*</italic> its dual. Recall that <italic>E*</italic> is not in general a Lie algebroid. We define a bilinear and symmetric form on the vector bundle <italic>E</italic> ⊕ <italic>E*</italic> 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.</p></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.

Dirac Structures on Banach Lie Algebroids

Faculty of Mathematics, ”Al. I. Cuza” University, Bd. Carol I, Iasi, 700508, Romania

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Dirac Structures on Banach Lie Algebroids"}

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...

Dirac Structures on Banach Lie Algebroids

Publié en ligne: 22 déc. 2015

Pages: 219 - 228

Reçu: 01 juil. 2013

Accepté: 01 oct. 2013

DOI: https://doi.org/10.2478/auom-2014-0060

Mots clés
Banach Lie algebroids, Dirac structures

© 2014 Vlad-Augustin Vulcu, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Dirac Structures on Banach Lie Algebroids

Publié en ligne: 22 déc. 2015

Pages: 219 - 228

Reçu: 01 juil. 2013

Accepté: 01 oct. 2013

DOI: https://doi.org/10.2478/auom-2014-0060

Mots clésBanach Lie algebroids, Dirac structures

© 2014 Vlad-Augustin Vulcu, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Mots clés
Banach Lie algebroids, Dirac structures