A standard Courant Lie algebroid of a manifold is a type of Courant algebroid constructed from the tangent bundle and cotangent bundle of . This is the principal algebraic structure studied in generalized complex geometry.
Recall from the discussion at Courant algebroid that there are the following two equivalent definitions of Courant algebroids:
In the first perspective a standard Courant algebroid of a manifold is the vector bundle – the fiberwise direct sum of the tangent bundle and the cotangent bundle – with
where denotes the Lie derivative of the 1-form by the vector field .
As a dg-manifold
As an dg-manifold a standard Courant algeebroid is is , the shifted cotangent bundle of the shifted tangent bundle,
where the differential (homological vector field) is on each local coordinate patch with coordinates
As a Lie 2-algebroid
We may read the above dg-algebra as the Chevalley–Eilenberg algebra of the Lie 2-algebroid , the specification of which entirely specifies the Lie 2-algebroid itself.
More on this in the discussion below.
As the Atiyah Lie 2-algebroid of a -gerbe
A standard Courant algebroid may be understood as being related to principal 2-bundles (-gerbes) as an Atiyah Lie algebroid is related to a -principal bundle.
Connections and generalized Riemannian metrics
Write for the standard Courant algebroid of the manifold . It comes canonically equipped with a projection down to the tangent Lie algebroid of :
of this morphism of Lie ∞-algebroids is often called a connection on . One may regard it as being special flat ∞-Lie algebroid valued differential form data on .
On base manifolds of the form sections of in the 1-category of Lie ∞-algebroids are in natural bijection with rank-2 tensor fields on , i.e. with sections .
The proof is straightforward and easy, but spelling it out in detail also serves to establish concepts and notation for the treatment of the Courant algebroid in terms of its Chevalley–Eilenberg algebra.
The Chevalley–Eilenberg algebra of the Lie 2-algebroid is the semifree dga whose underlying algebra is the Grassmann algebra
where the generators and are in degree 1 and the in degree 2, equipped with the differential that is defined on generators by
where are the canonical coordinate functions on .
The Chevalley–Eilenberg algebra of the tangent Lie algebroid is the deRham complex
The morphism is given by the dg-algebra morphism
that is the identity on and identifies the with the deRham differentials of the standard coordinate functions
A section is accordingly a dg-algebra morphism
Being a section, it has to be the identity on and send .
The image of the generators , being of degree 1, must be a linear combination over of the degree-1 elements in , i.e. must be 1-forms on . This defines the rank-2 tensor in question by
For this assignment to qualify as part of a morphism of dg-algebras, it has in addition to be compatible with the differential. The condition is that for all we have the equality in the bottom right corner of
This uniquely fixes the image under of the generators and the differential is respected. So, indeed, the section is specified by the tensor and every such tensor gives rise to a section.
The rank-2 tensor appearing in the above may be uniquely writtes as sum of a symmetric and a skew-symmetric rank-2 tensor and
If the symmetric part happens to be non-degenerate, it may be regarded as a (possibly pseudo-)Riemannian metric. In this case the combination is called a generalized Riemannian metric in generalized complex geometry.
Canonical -Lie algebroid 3-cocycle
The standard Courant albebroid is canonically equipped with the Lie ∞-algebroid 3-cocycle that is on a local patch given by
Morphisms between standard Courant algebroids
In the 1-category of Lie ∞-algebroids, automorphisms of the standard Courant algebroid of a cartesian space, , that
come from (…say this more precisely…) rank-2 tensors such that the skew symmetric part is a closed 2-form, .
With the same kind of reasoning as above, we find that the action on the generators and is of the form
For the 3-cocycle to be preserved, we need that
The description of the standard Courant algebroid in its incarnation as an dg-manifold is given for instance in section 5 of