Zavadskij modules over cluster-tilted algebras of type
Zavadskij modules are uniserial tame modules. They arose from interactions between the poset representation theory and the classification of general orders. The main problem is to characterize Zavadskij modules over a finite-dimensional algebra $ A $. In this setting, we prove that the indecomposabl...
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| Format: | Article |
| Language: | English |
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AIMS Press
2022-07-01
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| Series: | Electronic Research Archive |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2022175?viewType=HTML |
| _version_ | 1811223804586754048 |
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| author | Agustín Moreno Cañadas Robinson-Julian Serna Isaías David Marín Gaviria |
| author_facet | Agustín Moreno Cañadas Robinson-Julian Serna Isaías David Marín Gaviria |
| author_sort | Agustín Moreno Cañadas |
| collection | DOAJ |
| description | Zavadskij modules are uniserial tame modules. They arose from interactions between the poset representation theory and the classification of general orders. The main problem is to characterize Zavadskij modules over a finite-dimensional algebra $ A $. In this setting, we prove that the indecomposable uniserial $ A $-modules with a mast of multiplicity one in each vertex are Zavadskij modules. Since the Zavadskij property carries over to direct summands, but it is not invariant under the formation of direct sums, we give a criterion to determine when the direct sum of indecomposable Zavadskij modules is again a Zavadskij module. In addition, we use the triangulations of the $ n+3 $-gon associated with the cluster-tilted algebra of type $ \mathbb{A}_{n} $ to give a formula for the number of indecomposable Zavadskij modules over any cluster-tilted algebra of type $ \mathbb{A}_{n} $. In this case, the formula gives the dimension of the cluster-tilted algebra. As an application, we discuss some integer sequences in the OEIS (The On-Line Encyclopedia of Integer Sequences) that allow us to enumerate indecomposable Zavadskij modules. |
| first_indexed | 2024-04-12T08:38:02Z |
| format | Article |
| id | doaj.art-02d0341d539d4779bd5c5959cb9ec721 |
| institution | Directory Open Access Journal |
| issn | 2688-1594 |
| language | English |
| last_indexed | 2024-04-12T08:38:02Z |
| publishDate | 2022-07-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj.art-02d0341d539d4779bd5c5959cb9ec7212022-12-22T03:39:57ZengAIMS PressElectronic Research Archive2688-15942022-07-013093435345110.3934/era.2022175Zavadskij modules over cluster-tilted algebras of type Agustín Moreno Cañadas0Robinson-Julian Serna1Isaías David Marín Gaviria 21. Departamento de Matemáticas, Universidad Nacional de Colombia, Edificio Yu Takeuchi 404, Kra 30 No 45-03, Bogotá 11001000, Colombia2. Escuela de Matemáticas y Estadística, Universidad Pedagógica y Tecnológica de Colombia, Avenida Central del Norte 39-115, Tunja 150003, Colombia2. Escuela de Matemáticas y Estadística, Universidad Pedagógica y Tecnológica de Colombia, Avenida Central del Norte 39-115, Tunja 150003, ColombiaZavadskij modules are uniserial tame modules. They arose from interactions between the poset representation theory and the classification of general orders. The main problem is to characterize Zavadskij modules over a finite-dimensional algebra $ A $. In this setting, we prove that the indecomposable uniserial $ A $-modules with a mast of multiplicity one in each vertex are Zavadskij modules. Since the Zavadskij property carries over to direct summands, but it is not invariant under the formation of direct sums, we give a criterion to determine when the direct sum of indecomposable Zavadskij modules is again a Zavadskij module. In addition, we use the triangulations of the $ n+3 $-gon associated with the cluster-tilted algebra of type $ \mathbb{A}_{n} $ to give a formula for the number of indecomposable Zavadskij modules over any cluster-tilted algebra of type $ \mathbb{A}_{n} $. In this case, the formula gives the dimension of the cluster-tilted algebra. As an application, we discuss some integer sequences in the OEIS (The On-Line Encyclopedia of Integer Sequences) that allow us to enumerate indecomposable Zavadskij modules.https://www.aimspress.com/article/doi/10.3934/era.2022175?viewType=HTMLalgorithm of differentiationcategorificationcluster-tilted algebra of type ainteger sequenceoeisquiver representationuniserial modulezavadskij module |
| spellingShingle | Agustín Moreno Cañadas Robinson-Julian Serna Isaías David Marín Gaviria Zavadskij modules over cluster-tilted algebras of type Electronic Research Archive algorithm of differentiation categorification cluster-tilted algebra of type a integer sequence oeis quiver representation uniserial module zavadskij module |
| title | Zavadskij modules over cluster-tilted algebras of type |
| title_full | Zavadskij modules over cluster-tilted algebras of type |
| title_fullStr | Zavadskij modules over cluster-tilted algebras of type |
| title_full_unstemmed | Zavadskij modules over cluster-tilted algebras of type |
| title_short | Zavadskij modules over cluster-tilted algebras of type |
| title_sort | zavadskij modules over cluster tilted algebras of type |
| topic | algorithm of differentiation categorification cluster-tilted algebra of type a integer sequence oeis quiver representation uniserial module zavadskij module |
| url | https://www.aimspress.com/article/doi/10.3934/era.2022175?viewType=HTML |
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