Year: 2026
Type: preprint
Abstract: Let $\Bbbk$ be an algebraically closed field of characteristic different from $2$. We realize the four-dimensional selfinjective algebra $Ω(1)$ as an admissible infinite cyclic orbit algebra of the repetitive Kronecker category. Thus an algebra long stated to be nonstandard is of Euclidean type in the orbit-algebra sense. We also identify a gap in an earlier proof of its nonstandardness and give an independent proof by determining all connected Galois covering sources of $Ω(1)$. Together, these results disprove the implication from Euclidean type to standardness in the equivalence explicitly stated by Bocian--Holm--Skowroński (2004) and Bocian--Skowroński (2005). They also contradict the standardness conclusion of the Proposition in \S2.1 of Skowroński (1989); its main theorem in \S1.5 assumes standardness and is not contradicted by this example.
Source: arXiv (Cornell University)
Author: Asashiba, Hideto
Institution: Osaka Central Advanced Mathematical Institute
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223901619