LA-Courant algebroids and their applications

Abstract

LA-Courant algebroids and their applications.

On LA-Courant Algebroids and Poisson Lie 2-Algebroids

“…The category of Lie 2-algebroid is equivalent via the correspondence described above to the category of VB-Courant algebroids [10,8]. Note that a morphism µ :…”

“…This paper surveys the author's recent equivalences of graded manifolds of degree 2 with metric double vector bundles [7] (see also [1]), of self-dual 2-representations with decomposed metric VB-algebroids [7] (using [4]), and of split Lie 2-algebroids with decomposed VB-Courant algebroids [8], [10]. We combine all those results to provide an alternative definition for Li-Bland's LA-Courant algebroids [10], which are equivalent to Poisson Lie 2-algebroids.…”

“…The induced split Lie 2-algebroid is then equivalent to a VB-Courant algebroid structure in this decomposition [8] and the induced split Poisson structure of degree −2 is equivalent to the decomposition of a (metric) VB-algebroid structure on the other side of the metric double vector bundle [7]. The compatibility of the graded Poisson structure and the Lie 2-algebroid structure is equivalent to the VB-Courant algebroid and the metric VB-algebroid defining together an LA-Courant algebroid (short for Lie algebroid Courant algebroid ) [10]. Hence, we find that a metric double vector bundle with a VB-Courant algebroid structure and a metric VB-algebroid structure define together an LA-Courant algebroid if and only if, in any decomposition, the induced split Lie 2-algebroid and the induced self-dual 2representation build a matched pair.…”

“…The original definition of an LA-Courant algebroid [10] is inspired from and as technical as Mackenzie's first definition of a double Lie algebroid [15]. In short, a VB-Courant algebroid with a linear Lie algebroid on its other side is an LA-Courant algebroid if a relation defined by the Lie algebroid structure in the tangent prolongation of the VB-Courant algebroid -the underlying geometric structure is a triple vector bundle -is a Dirac structure in this tangent prolongation.…”

“…However, working out the equations directly from Li-Bland's LA-Courant algebroid condition is very long and technical (see the Appendix B of [5]). In this paper, we prefer therefore using Li-Bland's equivalence [10] in order to find the characterisation of an LA-Courant algebroid via the matched pair. We provide so a better (more handy) definition of LA-Courant algebroids.…”