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

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 $B(U)=M(U{)}^{\mathrm{\prime }}$. For every finite graded rank vector $h$, we prove that the moduli stack of such algebras is equivalent to the quotient stack $[{\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}^{\mathrm{g}\mathrm{r}}h(B(U))\mathrm{/}\mathrm{G}\mathrm{L}(U)]$. At a quotient $B(U)\to C$ with kernel $N$, its tangent complex is the two-term complex from $\mathrm{E}\mathrm{n}\mathrm{d}(U)$ to $\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{S}\mathrm{y}\mathrm{m}(U)(N,C{)}_0$. For every algebra $\mathfrak{g}$ in this stack, restriction to ${\mathrm{\Lambda }}^2{\mathfrak{g}}^{\mathrm{\prime }}$ induces a defect map on $H_0^2(\mathfrak{g};\mathfrak{g})$, 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 $\dim H_0^2=11$ and $H_0^3=0$. 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.