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 A₀ of primitive idempotents has the property that $\mathit{eAe}=\mathbb{k}$ for all $e\in A_0.$ Let n be a nonzero integer, and $\varphi$ and $\psi$ two automorphisms of the repetitive category $\widehat{A}$ of A with jump n (namely, they send $A^{[0]}$ to $A^{[n]},$ where $A^{[i]}$ is the i-th copy of A in $\widehat{A}$ 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 $\widehat{A}/〈\varphi 〉$ and $\widehat{A}/〈\psi 〉$ are isomorphic as $\mathbb{Z}$-graded categories.

Keywords:

• Algebra isomorphisms
• graded categories
• group actions
• orbit categories
• repetitive categories

2010 Mathematics Subject Classification:

• 18D05
• 16W22
• 16W50

Acknowledgments

We would like to thank Junichi Miyachi for asking us about the existence of examples that gives a negative solution to the conjecture above, Manuel Saorín for sending us his unpublished paper [11] (the published version of which is [10]) and Steffen Koenig for informing us the proof of Lemmas 5.1 and 5.2. Finally we also would like to thank the referee for helpful suggestions about relationships between this paper and the work of [7], which relates the main result with the inner automorphisms of algebras (see Lemma 4.3).

Notes

1 When B is regarded as a category by fixing a basic set of primitive idempotents of B as the set of objects, these automorphisms are not assumed to send objects of B to objects of B.

2 Note that if ${\varphi }_0^{-1}{\psi }_0=\xi ({\rho }_0),$ then $c_{x,y}$ can be given by $c_{x,y}=p{({\rho }_0)}_{x,y}$ because $\xi ({\rho }_0)(a)=q(p({\rho }_0))(a)=p{({\rho }_0)}_{x,y}a$ for all $x,y\in Q_0$ and $a\in A(x,y).$

Additional information

Funding

This work is partially supported by Grant-in-Aid for Scientific Research (grant nos. 25610003 and 25287001) from JSPS (Japan Society for the Promotion of Science 10.13039/501100001691), and by JST (Japan Science and Technology Agency), CREST Mathematics (10.13039/50110000338215656429). Michio Yoshiwaki was partially supported by Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849).