Automorphisms of locally string algebras
Sarafina Ford
TL;DR
This work provides a structural decomposition of automorphisms for (locally) string algebras, showing that vertex‑permuting automorphisms split into a graded part and a composition of exponential automorphisms of types I and II, with inner automorphisms capturing the residual action. Extending beyond finite dimensionality, the authors establish the same kind of decomposition for locally string algebras and demonstrate that Aut(A) can be understood via the fixed subgroups Aut_0(A), grAut(A), and Inn^{*}(A). A key technique is embedding locally gentle algebras with a unique infinite maximal path into matrix polynomial rings to reduce automorphism questions to inner automorphisms, while explicit derivation‑driven (exponential) automorphisms illuminate how arrows and maximal paths can be independently perturbed. The paper provides detailed quotient descriptions of Aut(A) modulo Inn^{*}(A) in the gentle and cycle cases and identifies normal subgroups (e.g., E(A), D(A)) controlling the exponential part. Overall, the results yield concrete, computable descriptions of automorphism groups for a broad class of string‑ and gentle‑type algebras, including infinite‑dimensional analogues, with explicit decompositions into graded, exponential, and inner components.
Abstract
It is known that automorphisms of finite-dimensional bound quiver algebras decompose into inner automorphisms and automorphisms which permute the vertices. In this paper, we show that for string algebras, automorphisms permuting vertices further decompose into a graded automorphism and a composition of certain types of exponential automorphisms. Moreover, the same decomposition applies to automorphisms of locally string algebras, which are an analogue of string algebras wherein the finite-dimensional condition is omitted.