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.

  • $\begingroup$ 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 $(L,[\cdot,\cdot])$. $\endgroup$ – YCor Feb 5 at 1:39
  • $\begingroup$ 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. $\endgroup$ – Denis T. Feb 5 at 1:46
  • $\begingroup$ 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 $i,j$. $\endgroup$ – YCor Feb 5 at 2:30
  • $\begingroup$ I do not understand why it should give the right answer anyways, sorry. $\endgroup$ – Denis T. Feb 5 at 8:12
  • $\begingroup$ Classification of nilpotent Lie algebras (even under some restrictions) is probably hopeless, with arbitrary families occuring as soon as the dimension gets large. $\endgroup$ – F. C. Feb 8 at 19:05

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