arXiv 1809.097602018-09math.RA

作用于子代数簇的高维代数的自同构群

Automorphism Groups of Finite-Dimensional Algebras Acting on Subalgebra Varieties

Alexander H. Sistko

备注：: Preprint version; Typo corrected, 23 pages, 7 figures

摘要：

设$k$是一个代数闭域，$B$是有$n := \dim_kB < \infty$的结合$k$ -代数.对于每个$1 \le m \le n$，$B$的所有$m$维子代数的集合携带一个投射簇的结构，我们称之为$\operatorname{AlgGr_m(B)}$ . $B$的所有$k$ -代数自同构的群$\operatorname{Aut}_k(B)$正则地作用在$\operatorname{AlgGr}_m(B)$上.本文研究了$\operatorname{AlgGr}_m(B)$的显式描述问题，并对其$\operatorname{Aut}_k(B)$ -轨道进行了分类.受有限维代数的极大子代数的最新结果的启发，我们计算了$B$为基时$\operatorname{AlgGr}_{n-1}(B)$的齐次消失理想，并明确地刻画了它的不可约分支.我们证明了在这种情况下，$\operatorname{AlgGr}_{n-1}(B)$是一个有限的联盟的$\operatorname{Aut}_k(B)$ -轨道，如果$B$是单项式或其Ext-的Schur，但构造了一类的例子来表明，这些条件是不必要的。

英文摘要：

Let $k$ be an algebraically-closed field, and $B$ a unital, associative $k$-algebra with $n := \dim_kB < \infty$. For each $1 \le m \le n$, the collection of all $m$-dimensional subalgebras of $B$ carries the structure of a projective variety, which we call $\operatorname{AlgGr_m(B)}$. The group $\operatorname{Aut}_k(B)$ of all $k$-algebra automorphisms of $B$ acts regularly on $\operatorname{AlgGr}_m(B)$. In this paper, we study the problem of explicitly describing $\operatorname{AlgGr}_m(B)$, and classifying its $\operatorname{Aut}_k(B)$-orbits. Inspired by recent results on maximal subalgebras of finite-dimensional algebras, we compute the homogeneous vanishing ideal of $\operatorname{AlgGr}_{n-1}(B)$ when $B$ is basic, and explictly describe its irreducible components. We show that in this case, $\operatorname{AlgGr}_{n-1}(B)$ is a finite union of $\operatorname{Aut}_k(B)$-orbits if $B$ is monomial or its Ext quiver is Schur, but construct a class of examples to show that these conditions are not necessary.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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