Formal Lie groupoids
For every Lie algebra or ∞-Lie algebra or ∞-Lie algebroid there is its Chevalley-Eilenberg algebra and its Weil algebra and a canonical dg-algebra morphism
Recall that a cocycle on is a closed element in . An invariant polynomial is a closed elements in that sits in the shifted copy .
This means that for , for the contraction derivation and the corresponding Lie derivative, we have in particular that an invariant polynomial is invariant in the sense that
For an ordinary Lie algebra, an invariant polynomial on is precisely a symmetric multilinear map on which is -invariant in the ordinary sense.
For an ∞-Lie algebroid (of finite type, i.e. degreewise of finite rank) with Chevalley-Eilenberg algebra
and Weil algebra
an invariant polynomial on is an elements with the property that
is a wedge product of generators in the shifted copy of , i.e.
or equivalently: for all and the contraction derivation, we have
it is closed in in that
or more generally its differential is again in the shifted copy.
We say an invariant polynomial is decomposable if it is the wedge product in of two invariant polynomials of non-vanishing degree.
Two invariant polynomials are horizontally equivalent if there is such that
Every decomposable invariant polynomial, def. 2, is horizontally equivalent to 0.
Let be a wedge product of two indecomposable polynomials. Then there exists a Chern-Simons element such that . By the assumption that is in non-vanishing degree and hence in it follows that
also is in
Therefore exhibits a horizontal equivalence .
Horizontal equivalence classes of invariant polynomials on form a graded vector space . There is a morphism of graded vector spaces
unique up to horizontal equivalence, that sends each horizontal equivalence class to a representative.
We write for the dg-algebra whose underlying graded algebra is the free graded algebra on the graded vector space , and whose differential is trivial.
Since invariant polynomials are closed, the inclusion of graded vector spaces from observation 2 induces an inclusion (monomorphism) of dg-algebras
On Lie algebras
For a Lie algebra, this definition of invariant polynomials is equivalent to more traditional ones.
To see this explicitly, let be a basis of and the corresponding basis of . Write for the structure constants of the Lie bracket in this basis.
Then for an element in the shifted generators, the condition that it is -closed is equivalent to
where the parentheses around indices denotes symmetrization, as usual, so that this is equivalent to
for all choice of indices. This is the component-version of the familiar invariance statement
for all .
On semisimple Lie algebras
See Killing form, string Lie 2-algebra.
On tangent Lie algebroids
For a smooth manifold, and invariant polynomial on the tangent Lie algebroid is precisely a closed differential form on .
On the string Lie 2-algebra
For a semisimple Lie algebra let be the canonical Lie algebra cocycle in degree 3, which is the one in transgression with the Killing form invariant polynomial .
Write for the corresponding string Lie 2-algebra. We have that the Chevalley-Eilenberg algebra is given by
and the Weil algebra is given by
where acts by degree shift isomorphism on unshifted generators.
It follows at once that every invariant polynomial
on the Lie algebra canonically identifies also with an invariant polynomial of the string Lie 2-algebra. But the differnce is that the Killing form is non-trivial as a polynomial on , but as a polynomial on becomes horizontally equivalent ,def. 3), to the trivial invariant polynomial.
Let be any Chern-Simons element for , hence an element such that
Then notice that by the above we have in that the differential of the new generator is equal to that of :
We on we can replace by and still get a Chern-Simons element for the Killing form:
But while is not in , the element is, by definition. Therefore is in that kernel, and hence exhibits a horizontal equivalence between and .
This is a special case of the more general statement below, about invariant polynomials on shifted central extensions.
For illustration purposes it is useful to consider the following variant of this example:
for the L-∞ algebra defined by the fact that its Chevalley-Eilenberg algebra is given by
where is a dual basis in degree 1 for some semisimple Lie algebra as above, and are generators in degree 2 and 3, respectively, and is the canonical Lie algebra cocycle in degree 3, as above.
It is easily seen that
The canonical morphism
given dually by sending
is a weak equivalence.
So the Lie 3-algebra is a kind of resolution of the ordinary Lie algebra . It is for instance of use in the presentation of twisted differential string structures, where the shifted piece in picks up the failure of -valued connections to lift to -2-connections.
The proof of the following proposition may be instructive for seeing how the definition of horizontal equivalence of invariant polynomials takes care of having the invariant polynomials of agree with those of .
There is an isomorphism
Notice that the Weil algebra of is given by
for new generators in degree 2, in degree 3 and in degree 4, coming with their Bianchi identities
For the following computations let be the structure constants of the Killing form, so that
and assume that is normalized such that
(if another normalization is chosen, then the corresponding factor will float around the following formulas without changing anything of the end result).
Now the indecomposable invariant polynomials are those of and one additional one: . This means that before deviding out horizontal equivalence on generators, the invariant polynomials of are not equal to those of , due to the superfluous generator .
But we do have the horizontal equivalence relation
where is any Chern-Simons element for , for instance
Notice that the homotopy here is indeed in : the component of not in that kernel is precisely . The above formula subtracts this offending summand and replaces it with the new generator , which by definition is in the kernel and whose image under is the image of under , plus the superfluous new generator of invariant polynomials.
Therefore in horizontal equivalence classes of invariant polynomials on the superfluous is identified with the Killing form , and hence the claim follows.
On symplectic Lie -algebroids
A symplectic Lie n-algebroid is an L-infinity algebroid that carries a binary and non-degeneraty invariant polynomial of grade . This is a generalization of the notion of symplectic form to which it reduces for .
On reductive Lie algebras
This appears for instance as (GHV, vol III, page 242, theorem I).
Role in -Chern-Weil theory
In (-)Chern-Weil theory the crucial role played by the invariant polynomials is their relation to ∞-Lie algebra cocycles. One may understand invariant invariant polynomials as extending under Lie integration -Lie algebra cocycles from cohomology to differential cohomology.
Transgression cocycles and Chern-Simons elements
(Chern-Simons elements and transgression cocycles)
Let be an ∞-Lie algebra. Since the cochain cohomology of the Weil algebra is trivial, for every invariant polynomial there is necessarily an element with
This we call a Chern-Simons element for .
This element will in general not sit entirely in the shifted copy. Its restriction
is a ∞-Lie algebra cocycle. We say this is in transgression with .
In total this construction yields a commuting diagram
where denotes the ∞-Lie algebra whose CE-algebra has a single generator in degree and vanishing differential, and where is the algebra of invariant polynomials of .
The element associated to an invariant polynomial by the above procedure is indeed a cocycle, and its cohomology class is independent of the choice of the element involved.
The procedure that assigns to 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
the element satisfies , hence that it is an ∞-Lie algebra cocycle;
any two different choices of lead to cocylces that are cohomologous.
This construction exhibits effectively the preimage of the connecting homomorphism in the cochain cohomology sequence induced by :
The dg-algebra of invariant polynomials is a sub-dg-algebra 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 :
Example. 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 On semisimple Lie algebras.
Chern-Simons and curvature characteristic forms
For a Lie n-alghebra, let be the ∞-Lie group obtained by Lie integration from it.
For a paracompact smooth manifold with good open cover whose Cech nerve we write , a cocycle for a -principal ∞-bundle on is cocycle with coefficients in the simplicial sheaf
We say an -connection on this is an extension to a cocycle with coefficients in the simplicial sheaf
The diagrams on the left encode those -valued forms on whose curvature vanishes on . One can show that one can always find a genuine -connection: one for which the curvatures have no leg along , in that they land in . For those the above diagram extends to
This defines the simplicial presheaf that classifies connections on ∞-bundles.
By pasting-postcomposition with the above diagrams for an invariant polynomial we obtain connections with values in
where in the bottom row we have the curvature characteristic forms coresponding to the connection, and in the middle the corresponding Chern-Simons forms.
More details for the moment at ∞-Chern-Weil theory introduction.
Invariant polynomials for Lie algebras of simple Lie groups are disussed in
A standard textbook account of the traditional theory is in volume III of
The notion of invariant polynomials of -algebras has been introduced in
An account in the more general context of Lie theory in cohesive (infinity,1)-toposes is in section 3.3.11 of