Something has been bothering me about Koszul duality lately. Well, technically many things have been bothering me about it, but here’s a particular thing that has been bothering me. Usual (homological) Koszul duality assigns to an augmented -dga the algebra . Since is central, is again an augmented -dga, and so makes sense. In many (all?) cases, (hence, duality).
I am interested in the case when is no longer central in . Then an action of on no longer is the same as an augmentation map, so we only assume we have a left -module structure on (where the contained copy of acts by mutliplication). We can still form in the same way; however, no longer contains , so it seems impossible to form .
However, I have reason to suspect that one should be able to form , and that it should often be quasi-isomorphic to . My evidence in this direction is a paper of Positsel’skii’s, “Nonhomogeneous Quadratic Duality and Curvature”, wherein he shows that for a very narrow class of algebras which both contain and act on , that can be recovered from .
An important case of this duality is for the ring of differential operators on a smooth curve. This has a canonical left action on , and so exists, and is in fact the deRham complex. So, a specific case of my question is: is there a sense in which the deRham complex of a space acts on , such that something like ?
More generally, one can replace with the universal enveloping algebra of any Lie algebroid over . can still be defined, and is the complex which computes the Lie algebroid cohomology.
Both of these examples do fall under Positsel’skii’s explicit duality. The problem is that the framework of this duality isn’t self-contained. It takes filtered algebras which aren’t too far from being quadratic algebras, and sends them to commutative dgas whose underlying graded algebra is quadratic. Then there is a different construction which takes quadratic commutative dgas to filtered algebras which are almost quadratic, and these two functors induce an equivalence of categories.
What I want, though, is for these two dualities to be facets of the same duality. Maybe its not possible, but I imagine that people have thought about this, at least in the specific case of Lie algebroids.