Given a smooth manifold , one can talk about , the ring of differential operators on the dga of differential forms. Equipping with a riemannian structure determines a Hodge star, and conjugation with the Hodge star is an involution of .

I am very curious about the fixed set of this involution, the differential operators which commute with the Hodge star. They came up when I was trying to find a universal property for the non-commutative Weil algebra (in the sense of Alexeev and Meinrenken). Given an invariant metric on G, and a metric-preserving action of G on a riemannian manifold M, I believe that the non-commutative Weil algebra maps into the sub algebra of which commutes with the Hodge star.

Since the above Koszul duality swaps and , I was hoping that it would map the self-dual differential operators to themselves, and maybe say something about why this algebra carries any topological significance.

]]>in the article of Kapranov “On DG modules over the de Rham complex..”

Indeed O is a de Rham module corresponding to D —

it might be easier to see by writing down the full

de Rham complex of D as a left D-module, and checking it

is quasiisomorphic to O.

The way he treats the duality also generalizes to Lie algebroids:

it makes life much easier to not start with D and de Rham,

or U(L) and Chevalley(L), but to look at the Rees algebra of D which

is now a graded algebra, and its Koszul dual which is the exterior

algebra of L^* adjoined the Chevalley differential as an

extra variable. One gets the duality you want by localizing

wrt the extra variable on both sides.

As for the general theory I think Quillen’s article

on rational homotopy theory is really inspiring –

he works out the commutative/Lie Koszul duality, but

without calling it such. The Lie algebroid/Chevalley

case is a relative version of this. Quillen’s

article also gives context as to why we’re taking bar/cobar

complexes in Koszul duality — it corresponds to looping

and delooping spaces (the Lie algebra side can be thought of

as the Lie algebra of a loop space – which is a group roughly speaking — and the commutative algebra side corresponds to the original space,

which is the delooping or classifying space of its loop space.)

In geometry this corresponds to the modern approach to deformation

theory – starting from a commutative algebra you calculate

a (dg) Lie algebra, which is its tangent space (complex),

while from a Lie algebra you calculate the classifying space

of its formal group, a.k.a. the space of solutions to the Maurer-Cartan equation.