Transitive Courant algebroids

Izu Vaisman

doi:10.1155/ijmms.2005.1737

Abstract

We express any Courant algebroid bracket by means of a metric connection, and construct a Courant algebroid structure on any orthogonal, Whitney sumE⊕CwhereEis a given Courant algebroid andCis a flat, pseudo-Euclidean vector bundle. Then, we establish the general expression of the bracket of a transitive Courant algebroid, that is, a Courant algebroid with a surjective anchor, and describe a class of transitive Courant algebroids which are Whitney sums of a Courant subalgebroid with neutral metric and Courant-like bracket and a pseudo-Euclidean vector bundle with a flat, metric connection. In particular, this class contains all the transitive Courant algebroids of minimal rank; for these, the flat term mentioned above is zero. The results extend to regular Courant algebroids, that is, Courant algebroids with a constant rank anchor. The paper ends with miscellaneous remarks and an appendix on Dirac linear spaces.

Highlights

We express any Courant algebroid bracket by means of a metric connection, and construct a Courant algebroid structure on any orthogonal, Whitney sum E ⊕ C where E is a given Courant algebroid and C is a flat, pseudo-Euclidean vector bundle
We establish the general expression of the bracket of a transitive Courant algebroid, that is, a Courant algebroid with a surjective anchor, and describe a class of transitive Courant algebroids which are Whitney sums of a Courant subalgebroid with neutral metric and Courant-like bracket and a pseudo-Euclidean vector bundle with a flat, metric connection
The results extend to regular Courant algebroids, that is, Courant algebroids with a constant rank anchor

Summary

The basics of Courant algebroids

The framework of this paper is the C∞-category. In the literature, there are two notions of a Courant algebroid, which include a skew-symmetric and a non-skew-symmetric bracket, respectively. Property (c) implies that the skew-symmetric part of a Courant algebroid product satisfies the property [e, ∂. A skew-symmetric Courant algebroid is a pseudo-Euclidean vector bundle (E → M,g) with an anchor morphism ρ : E → TM and a skew-symmetric bracket [·,·] on ΓE such that properties (i)–(v) of Proposition 1.5 hold. Conditions (i), (iii), and (v) suffice in the definition of a skew-symmetric Courant algebroid. Using (1.15) it is easy to check that any pre-Courant algebroid satisfies property (iii) of Proposition 1.5 if at least one of the arguments ea ∈ im ∂ (a = 1, 2, 3) (the cross-sections of the subbundle im ∂ are locally spanned over C∞(M) by cross-sections of the form ∂ f ( f ∈ C∞(M))). Any g-isotropic subalgebroid (i.e., a vector subbundle that is closed by brackets) of a skew-symmetric Courant algebroid is a Lie algebroid

Courant brackets and metric connections

Transitive and regular Courant algebroids