# Moduli space of nilpotent Lie algebras

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.

• It would be natural to fix the given Carnot grading on $L$, consider the set $B_L$ of bilinear maps $b:L\otimes L\to L$ such that $b(L^i\otimes L^j)\subset L^{i+j+1}$ for all $i,j$ such that $[\cdot,\cdot]+b$ is a Lie algebra law on $L$. And then mod out by the action of the graded automorphism group of $(L,[\cdot,\cdot])$. – YCor Feb 5 at 1:39
• But if we consider only +1 degree part of bilinear maps, this will give only a germ of $M$ at $L$, right? And considering all positive degrees breaks this approach, because quotening by homogenous automorphisms does not give anything meaningful in that case. – Denis T. Feb 5 at 1:46
• Sorry, I meant this where $L^i$ is the lower central series. In terms of grading, it means $b(L_i\otimes L_j)\subset \bigoplus_{k>i+j}L_k$ for all $i,j$. – YCor Feb 5 at 2:30
• I do not understand why it should give the right answer anyways, sorry. – Denis T. Feb 5 at 8:12
• Classification of nilpotent Lie algebras (even under some restrictions) is probably hopeless, with arbitrary families occuring as soon as the dimension gets large. – F. C. Feb 8 at 19:05