A nonstandard selfinjective algebra of Euclidean type
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.
Related
- A Frankel type theorem in Euclidean and hyperbolic spaces
- On a derived equivalence classification of weakly symmetric algebras of Euclidean type
- A Littlewood-Type Theorem for Harmonic Functions in Euclidean Half-Spaces
- Hermitian Clifford Analysis on Euclidean Domains with Smooth and Fractal Boundaries: A Survey of Matrix Cauchy and Hilbert Transforms, Boundary Value Problems, Helmholtz-Type Dirac Operators and Duality
- The Extension groups of one K-type representations of graded Hecke algebra
BibTeX
@misc{asashiba2026nonstandard,
title = {{A nonstandard selfinjective algebra of Euclidean type}},
author = {Hideto Asashiba},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34119},
eprint = {2610.34119},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34119},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A nonstandard selfinjective algebra of Euclidean type
AU - Hideto Asashiba
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34119
UR - https://arxiv.org/abs/2610.34119
AB - 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.
ER -