Quot-Stack Moduli and Transverse Deformations of Graded Metabelian Lie Algebras

Abstract

We identify the graded metabelian locus inside the deformation theory of positively graded Lie algebras. Let $M(U)$ be the free metabelian Lie algebra on $U$ and $B(U)=M(U)'$. For every finite graded rank vector $h$, we prove that the moduli stack of such algebras is equivalent to the quotient stack $[\mathrm{Quot}^{\mathrm{gr}}h(B(U))/\mathrm{GL}(U)]$. At a quotient $B(U)\to C$ with kernel $N$, its tangent complex is the two-term complex from $\mathrm{End}(U)$ to $\mathrm{Hom}{\mathrm{Sym}(U)}(N,C)_0$. For every algebra $\mathfrak{g}$ in this stack, restriction to $\Lambda^2\mathfrak{g}'$ induces a defect map on $H^2_0(\mathfrak{g};\mathfrak{g})$, and we prove that its kernel is $H^0$ of the Quot-stack tangent complex. Thus a first-order deformation is tangent to the metabelian locus exactly when its derived--derived restriction vanishes. We also recover the inverse-system module degree by degree from intrinsic lower-central tensors; on the level locus its terminal tensor suffices. For the $14$-dimensional algebra attached to a regular pencil of binary quartics, exact computation gives $\dim H^2_0=11$ and $H^3_0=0$. Its effective miniversal graded deformation germ is formally smooth of dimension $11$, while its metabelian subgerm is formally smooth of dimension $3$. Terminal restriction identifies the $8$-dimensional normal space with derived--derived brackets. Using the pencil's classical $V_4$-symmetry, we determine its induced representation on the graded tangent space and show that four negative-weight primary obstruction maps are surjective.