自由亚交换李代数的层级商：逆系统与分次变形
Quot-Stack Moduli and Transverse Deformations of Graded Metabelian Lie Algebras
AI总结：
研究自由亚交换李代数的层级商，通过逆系统描述一度生成的有限维正分次亚交换李代数，给出相关构造及性质，包括轨道分类、自同构群等，还涉及不同维度情况及变形基等内容。
AI中文摘要：
设\(M(U)\)是有限维向量空间\(U\)上的自由亚交换李代数，\(B(U)=M(U)'\)是其在\(\text{Sym}(U)\)上的导出模。我们通过\(B(U)\)的逆系统描述了一度生成的有限维正分次亚交换李代数。\(B(U)\)的受限分次对偶与闭除幂多项式\(2\)-形式的模等同，所得的商构造定义了广群的等价。在层级轨迹上，一个内在的终端换位子张量恢复了顶部逆系统空间，给出了轨道分类和终端标记实现的尖锐 catalecticant 下界。当\(\dim U = 2\)时，模\(B(U)\)是循环的，该构造成为二元形式子空间与有限维、双生成层级亚交换李代数之间的行列式扭曲等价。我们还确定了它们的分次和全自同构群。相比之下，在秩为三时，我们展示了产生相同李商的不同有限余长度标量理想，表明标量理想不能忠实地参数化高阶构造。对于与二元四次式的正则铅笔相关的\(14\)维代数，精确的有理计算给出了伴随\(H^2\)和\(H^3\)的完整卡诺权重分解，其中\(\dim H^2_0 = 11\)且\(H^3_0 = 0\)。因此，分次变形基是维度为\(11\)的光滑基；其亚交换轨迹是一个光滑的\(3\)维子芽，终端限制将\(8\)维法空间与新的导出 - 导出括号等同。我们确定了剩余的\(V_4\)作用及其不变环以及终端秩层。权重为零的变形无障碍，而四个负权重的主障碍配对是满射的。
英文摘要：
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 $Λ^2\mathfrak{g}'$ 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.
