On Isoclasses of Maximal Subalgebras Determined by Automorphisms

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