On Isoclasses of Maximal Subalgebras Determined by Automorphisms

TL;DR

This paper studies the classification of maximal subalgebras of path algebras of type A Dynkin quivers using automorphism group actions on a projective variety of subalgebras, revealing when isomorphism coincides with orbit equivalence.

Contribution

It establishes that for type A Dynkin quivers, isomorphic maximal subalgebras with connected Ext quivers are precisely those in the same automorphism group orbit.

Findings

Isomorphism of maximal subalgebras corresponds to automorphism group orbits.

Maximal subalgebras with connected Ext quivers are classified by these orbits.

Provides a geometric perspective on subalgebra classification.

Abstract

Let $k$ be an algebraically-closed field, and let $B = kQ/I$ be a basic, finite-dimensional associative $k$-algebra with $n := \dim_kB < \infty$. Previous work shows that the collection of maximal subalgebras of $B$ carries the structure of a projective variety, denoted by $\operatorname{msa} (Q)$, which only depends on the underlying quiver $Q$ of $B$. The automorphism group $\operatorname{Aut}_k(B)$ acts regularly on $\operatorname{msa} (Q)$. Since $\operatorname{msa} (Q)$ does not depend on the admissible ideal $I$, it is not necessarily easy to tell when two points of $\operatorname{msa} (Q)$ actually correspond to isomorphic subalgebras of $B$. One way to gain insight into this problem is to study $\operatorname{Aut}_k(B)$-orbits of $\operatorname{msa} (Q)$, and attempt to understand how isoclasses of maximal subalgebras decompose as unions of $\operatorname{Aut}_k(B)$-orbits. This paper investigates the problem for $B = kQ$, where $Q$ is a type $\mathbb{A}$ Dynkin quiver. We show that for such $B$, two maximal subalgebras with connected Ext quivers are isomorphic if and only if they lie in the same $\operatorname{Aut}_k(B)$-orbit of $\operatorname{msa} (Q)$.