A nonstandard selfinjective algebra of Euclidean type

Hideto Asashiba

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arXiv · 2610.34119

A nonstandard selfinjective algebra of Euclidean type

Hideto Asashiba

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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.

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Hideto Asashiba. 2026-10-25. A nonstandard selfinjective algebra of Euclidean type. https://arxiv.org/abs/2610.34119

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