Automorphism Groups of Finite-Dimensional Algebras Acting on Subalgebra Varieties

TL;DR

This paper investigates the structure and automorphism group actions on varieties of subalgebras of finite-dimensional algebras, providing explicit descriptions and classifications of orbits, especially for subalgebras of codimension one.

Contribution

It explicitly describes the subvariety of codimension-one subalgebras for basic algebras and classifies their automorphism group orbits, extending understanding of algebra automorphisms on subalgebra varieties.

Findings

Computed the homogeneous vanishing ideal of subalgebra varieties for basic algebras.

Described irreducible components of these subvarieties.

Identified conditions under which the subvariety is a finite union of automorphism orbits.

Abstract

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.