零维单项式代数的自同构群的结构
The structure of automorphism groups of zero-dimensional monomial algebras
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中文总结 AI 辅助
本文研究零维单项式代数的自同构群结构，给出生成元算法，证明分量群任意性，并应用于 Anick 定理的新证明。
中文摘要 AI 辅助
设 $A$ 是特征为零的代数闭域上的零维单项式代数，即多项式环关于一个单项式理想的有限维商。其自同构群 $G$ 是一个线性代数群，通过 $A$ 的齐次幂零导子来描述。我们详细分析 $G$ 的结构。其单位分支 $G^0$ 是其幂幺根与一个同构于一般线性群乘积的约化子群的半直积，并且对于每个根次数，我们刻画了相关导子何时产生一个加法根子群，并确定其维数。利用这些导子的李括号，我们随后给出一个显式算法，从理想的极小单项式生成元出发，生成一个根子群族，该族与一个极大环面一起生成 $G^0$。在一般情况下，这样的族是极小的。我们还证明了分量群 $G/G^0$ 可以是任意的：每个有限群都作为某个零维单项式代数的自同构群的分量群出现。最后，我们将这些结果应用于代数 $\mathbf{k}[\mathbf{x}]/\mathfrak{m}^d$，表明由极大环面和外部根子群生成的子群恰好是雅可比行列式为常数的自同构子群，并由此推导出关于 $\mathbf{k}[\mathbf{x}]$ 的 tame 自同构稠密性的 Anick 定理的一个新证明。
英文摘要
Let $A$ be a zero-dimensional monomial algebra over an algebraically closed field of characteristic zero, that is, a finite-dimensional quotient of a polynomial ring by a monomial ideal. Its automorphism group $G$ is a linear algebraic group, described through the homogeneous nilpotent derivations of $A$. We analyze the structure of $G$ in detail. Its identity component $G^0$ is a semidirect product of its unipotent radical and a reductive subgroup isomorphic to a product of general linear groups, and for each root degree we characterize when the associated derivations give rise to an additive root subgroup, and determine its dimension. Using the Lie brackets of these derivations, we then give an explicit algorithm that produces, out of the minimal monomial generators of the ideal, a family of root subgroups generating $G^0$ together with a maximal torus. Such a family is minimal in the generic case. We also show that the component group $G/G^0$ can be arbitrary: every finite group arises as the component group of the automorphism group of some zero-dimensional monomial algebra. Finally, we apply these results to the algebras $\mathbf{k}[\mathbf{x}]/\mathfrak{m}^d$, showing that the subgroup generated by a maximal torus and the outer root subgroups is exactly the subgroup of automorphisms with constant Jacobian determinant, and we deduce from this a new proof of Anick's theorem on the density of the tame automorphisms of $\mathbf{k}[\mathbf{x}]$.
发表机构
- Universidad de La Serena（拉塞雷纳大学）
- Universidad de Chile（智利大学）
- Universidad de Valparaíso（瓦尔帕莱索大学）
机构由 AI 辅助整理，请以论文原文为准。
