A symplectic -Lie algebroid is a pair consisting of an ∞-Lie algebroid and a non-degenrate binary invariant polynomial on .
This is equivalent to the definition of n-symplectic manifold (or symplectic NQ-supermanifold) as found in the literature. Here we prefer the fully -Lie theoretic language over that of NQ-supermanifolds only because it seems to be more suggestive of the structure of the problem.
An -symplectic ∞-Lie algebroid is a pair consisting of an -Lie algebroid and a nondegenerate binary invariant polynomial of degree .
The Poisson tensor of an -symplectic -Lie algebroid is a cocycle that is in transgression with .
The corresponding L-∞-algebra of sections is the -algebra of sections of the -Lie algebra obtained by co-killing this cocycle, i.e the homotopy fiber
The following paragraph recalls the meaning of an -Lie algebroid cocycle being in transgression with an invariant polynomial.
We recall the procedure by which to an ∞-Lie algebroid invariant polynomial we associate an ∞-Lie algebroid cocycle that is in transgression with .
The dg-algebra of invariant polynomials is a sub-dg-alghebra of the kernel of the morphism from the Weil algebra to the Chevalley-Eilenberg algebra of
From the short exact sequence
we obtain the long exact sequence in cohomology
We say that is in transgression with if their classes map to each other under the connecting homomorphism :
The following spells out in detail how one finds to a given invariant polynomial the cocycle that it is in transgression with.
We first regard the invariant polynomial as an element of the Weil algebra under the inclusion , where, by the very definiton of invariant polynomials, it is closed: .
then we find an element with the property that . This is guranteed to exist because has trivial cohomology.
then we send this element along the restriction map to an elemeent we call .
The procedure is illustarted by the following diagram
From the fact that all morphisms involved respect the differential and from the fact that the image of in vanishes it follows that
this element satisfies , hence that it is an -Lie algebroid cocycle.
any two different choices of lead to cocylces that are cohomologous.
We say is a cocycle in transgression with . We may call here a Chern-Simons element of . Because for any collection of ∞-Lie algebroid valued differential forms coming dually from a dg-morphism the image of will be a curvature characteristic form and the image its corresponding Chern-Simons form.
In the case where is an ordinary semisimple Lie algebra, this reduces to the ordinary study of ordinary Chern-Simons 3-forms associated with -valued 1-forms. This is described in the section Semisimple Lie algebras .
We spell out examples of symplectic -Lie algebroids in order or increasing .
The simplest example that makes closest contact to well familiar objects and constructions is that where is an ordinary semisimple Lie algebra and its ordinary binary invariant polynomial. Since here is of degree 4 in this is really a degenerate example of a symplectic Lie 2-algebroid and is accordingly listed in that subsection. The reader in need of pedagogical examples should start with that example and then jump back to the section on Poisson manifolds.
When the binary is of degree 2 it must be locally the wedge product of two elements of degree 1. Since these must be in the shifted copy of the Weil algebra they must come from element in degree 0 of . For to be non-degenerate, this implies that may not have generators of higher degree. Hence is a Lie 0-algebroid over a manifold , so it is just that manifold itself. Its Chevalley-Eilenberg algebra is the algebra of smooth functions on
The Weil algebra of is
the deRham algebra of . A degree 2-invariant polynomial on is therefore a non-degenerate closed 2-form , a symplectic 2-form.
A symplectic manifold, being a pair
consisting of a smooth manifold and a symplectic 2-form , is a symplectic Lie 0-algebroid.
Poisson manifolds – 1-symplectic manifolds
For a Poisson manifold with Poisson bivector the Chevalley-Eilenberg algebra of the corresponding Poisson Lie algebroid
is that of multi-vector fields on , equipped with the differential given by the Schouten bracket.
Let’s be traditional and work locally in coordinates so that we can recognize the following expressions as familiar friends from the literature.
So then is generated from degree 0 elements and degree 1 elements . The differential is
The Poisson tensor is and that this is a Lie algebroid cocycle is the fact that
By definition the Weil algebra is generated from the , the and their shifted partners and . The differential here is
We claim the invariant polynomial that is in transgression with the cocycle is
One checks directly that the element
is a Chern-Simons transgression element for and ,
i.e. . The restriction of to is evidently the Poisson tensor .
For a Poisson manifold with Poisson tensor , the pair
consisting of the Poisson lie algebroid and of the invariant polynomial that is in transgression with its canonical 2-cocycle (the Poisson tensor) is a symplectic Lie algebroid.
Courant algebroids - 2-symplectic manifolds
see e.g. Roytenberg’s discussion of Courant algebroids and proceed entirely analogously, with replaced by Dmitry’s
For a Courant algebroid over a base manifold the pair
for the invariant polynomial that is in transgression with the canonical 3-cocycle is a symplectic Lie 2-algebroid.
Semisimple Lie algebras – 2-symplectic points
The most classical and possibly most familiar case of all the examples here is possibly that of a Courant algebroid whose base space is a point. As a symplectic 2-Lie algebroid this is nothing but
an ordinary Lie algebra
equipped with a non-degenrate binary invariant polynomial .
For instance a semisimple Lie algebra equipped with its Killing form.
In that case one finds that the cocycle in transgression with is the canonical 3-cocycle
of . Co-killing this cocycle yields the String Lie 2-algebra , which is at the same time the Lie 2-algebra of sections of this example.
It may be worthwhile to spell this example out in terms of components of a chosen basis.
So let be a basis for the vector space underlying the Lie algebra , which we assume for simplicity to be finite-dimensional. Let be a corresponding dual basis of the dual vector space .
Let be the structure constants of the Lie bracket in this basis, given by
The Chevalley-Eilenberg algebra of is
is the Grassmann algebra of . It is the graded-commutative -algebra whose elements are linear combinations of wedge product expressions , , and so onw, where is of degree 1.
The differential is fully specified by specifying it on the generators , because from there it extends uniquely by the graded Leibnitz rule
to the rest of the algebra. On the differential is simply the linear-dual of the Lie bracket
In terms of our chosen basis with the structure constants as above this means that
The minus sign here is just convention and could be omitted without changing anything of the discussion. Its purpose is to make these expressions come out nicer when we pass next to the Weil algebra.
That this squares to 0
is equivalent to the vaishing of all the components
where the square brackets indicate antisymmetrization over the corresponding indices. This is indeed precisely equivalent to the Jacobi identity satisfied by the Lie bracket , as one can check.
Write now for the components of the invariant polynomial on in that
It is a standard fact in the general theory of Lie algebra cohomology that the element
and hence a Lie algebra 3-cocycle (for which the constant is irrelevant). We can easily see this fact systematically by passing now to the Weil algebra.
The Weil algebra is the dg-algebra
where is a copy of but shifted up in degree by one. We have
Write for the basis of obatained from the basis of .
If we write for the degree 1 morphism of graded vector spaces that is the identity on the underlying un-graded vector spaces, then
The differential on the Weil algebra is again fixed by its action generators. Restricted to the original generators we have
In terms of our chosen basis this reads
One finds that from this and using the action of is already fixed. It is
In terms of our basis this is
Now, the invariant polynomial is to be regarded as an element in the shifted part of which in terms of our basis reads
Notice that since the are in degree 2, we have .
The statement that this is a closed element of is now
and hence that all the components
vanish. Indeed, we see that this means that is completely antisymmetric in all its indices.
Now we want to find the Chern-Simons element corresponding to . There is a systematic formula for finding that, but in the simple case at hand here one sees that it has to be of the form
Finally, the restriction of along the projection is simply obtained by setting all the shifted generators to 0. So this does yield as the transgression cocycle
consisting of a semisimple Lie algebra and the invariant polynomial given by its invariant bilinear form (with some choice of normalization) is a symplectic Lie 2-algebra.
For reference on the notion of n-symplectic manifold in general and Poisson Lie algebroid and Courant algebroid in particular, please see these entries.
A discussion of cocycles and invariant polynomials in transgression on an -Lie algebroid is in
- Sati, Schreiber, Stasheff, -connections in: B. Fauser, J. Tolksdorf, E. Zeidler (eds.) Recent Progress in Quantum Field Theory – Competetive Models Birkhäuser, Basel (2009) (arXiv)