ag.algebraic geometry – moduli space of Nilepents Lie algebras

Repair a Nilpotent Lie algebra $ L $ over a char 0 field $ k $ that's of course graded, me. e. isomorphic to graduated algebra $ bar L $ in connection with the lower central filtration.

I am interested in a reasonable description of $ M subset Hom (L otimes L, L) $ consisting of algebras $ M $ with an algebra isomorphism $ phi: bar M to L $, Adequate description includes the action of $ GL (L) $ and some constant compactification.

Perhaps someone knows much more and has already found a way to describe slices of nilpotent algebras for which the PBW morphism is, in a sense, a deformation of the Coalgebra map – as in the Lefevre-Hasegawa thesis; I think this remark needs some elaboration, which is best suited as a separate question. Therefore, references to any articles about this type of "Lie algebra discs with connection" are welcome.