Fix a nilpotent Lie algebra $L$ over some char 0 field $k$ which is naturally graded, i. e. isomorphic to graded algebra $\bar L$ associated to lower central filtration.

I'm interested in some reasonable description of $M \subset Hom(L \otimes L, L)$ consisting of algebras $M$ with an algebra isomorphism $\phi: \bar M \to L$. Reasonable description includes action of $GL(L)$ and some invariant compactification.

Maybe someone knows a lot more and have already found a way to describe sheaves of nilpotent algebras for which PBW morphism is a deformation of coalgebra map in some sense — like in Lefevre-Hasegawa thesis; I guess this remark needs some elaboration, which is best suited as separate question. So, references to any papers about this kind of "Lie algebra sheaves with connection" are welcome.