On isomorphisms of generalized multifold extensions of algebras without nonzero oriented cycles

Abstract

Assume that a basic algebra $A$ over an algebraically closed field $\mathbb{k}$ with a basic set $A_0$ of primitive idempotents has the property that $eAe=\mathbb{k}$ for all $e\in A_0$. Let $n$ be a nonzero integer, and $\varphi$ and $\psi$ two automorphisms of the repetitive category $\hat{A}$ of $A$ with jump $n$ (namely, they send $A^{[0]}$ to $A^{[n]}$, where $A^{[i]}$ is the $i$-th copy of $A$ in $\hat{A}$ for all $i\in \mathbb{Z}$). If $\varphi$ and $\psi$ coincide on the objects and if there exists a map $\rho :A_0\to \mathbb{k}$ such that ${\rho }_0(y){\varphi }_0(a)={\psi }_0(a){\rho }_0(x)$ for all morphisms $a\in A(x,y)$, then the orbit categories $\hat{A}/⟨\varphi ⟩$ and $\hat{A}/⟨\psi ⟩$ are isomorphic as $\mathbb{Z}$-graded categories.

21 pages