On Isoclasses of Maximal Subalgebras Determined by Automorphisms

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 $\operatorname{msa} (Q)$.