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.
Highlights
In the early nineties Courant algebroids appeared in Poisson geometry as a convenient framework for the study of constrained Hamiltonian systems [1]
We show that the symplectic structure of the symplectic Lie 2-algebroid that corresponds to a Courant algebroid could rather be understood as a structure of tangent space, just like the canonical symplectic structure on a cotangent bundle T ∗M is just the dual Poisson structure to the standard tangent Lie algebroid T M → M, which happens to be nondegenerate
Since LA-Courant algebroids are equivalent to Poisson Lie 2-algebroids [14], 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 2-representation build a matched pair
Summary
In the early nineties Courant algebroids appeared in Poisson geometry as a convenient framework for the study of constrained Hamiltonian systems [1]. They are at the base of the definition of generalised complex geometry [6, 7]. The graded geometric description of Courant algebroids as symplectic Lie 2-algebroids are due to
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