Abstract
A Nakayama algebra with almost separate relations is one where the overlap between any pair of relations is at most one arrow. In this paper we give a derived equivalence between such Nakayama algebras and path algebras of quivers of a special form known as quipu quivers. Furthermore, we show how this derived equivalence can be used to produce a complete classification of linear Nakayama algebras with almost separate relations. As an application, we include a list of the derived equivalence classes of all Nakayama algebras of length \(\le 8\) with almost separate relations.
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25 January 2024
A Correction to this paper has been published: https://doi.org/10.1007/s10468-024-10252-2
References
Amiot, C., Oppermann, S.: Algebras of acyclic cluster type: Tree type and type Ã. In: Nagoya Mathematical Journal 211, 1–50 (2013)
Caldero, P., Keller, B.: From triangulated categories to cluster algebras II. In: Annales scientifiques de l’École Normale Supérieure Ser. 4, 39.6, 983–1009 (2006)
Fosse, D.: A combinatorial procedure for tilting mutation. (2021). arXiv:2112.08129
Ghebleh, M., Kanso, A., Stevanović D.: Open quipus with the same Wiener index as their quadratic line graph. In: Applied Mathematics and Computation 281, 130–136 (2016)
Lan, J., Shi, L.: Graphs of order n and diameter \(2(n - 1)/3\) minimizing the spectral radius. In: Linear Algebra and its Applications 486, 219–233 (2015)
Woo, R., Neumaier, A.: On Graphs Whose Spectral Radius is Bounded by \(\frac{3}{2}\sqrt{2}\) In: Graphs and Combinatorics 23 (2007)
Acknowledgements
This paper forms part of the authors PhD thesis, written under the supervision of Professor Steffen Oppermann. The author thanks Professor Oppermann for helpful and inspiring discussions.
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Open access funding provided by NTNU Norwegian University of Science and Technology (incl St. Olavs Hospital - Trondheim University Hospital)
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Fosse, D. Quipu Quivers and Nakayama Algebras with Almost Separate Relations. Algebr Represent Theor 27, 995–1010 (2024). https://doi.org/10.1007/s10468-023-10247-5
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DOI: https://doi.org/10.1007/s10468-023-10247-5