Zavadskij modules over cluster-tilted algebras of type $ \mathbb{A} $

Agustín Moreno Cañadas; Robinson-Julian Serna; Isaías David Marín Gaviria; Agustín Moreno Cañadas; Robinson-Julian Serna; Isaías David Marín Gaviria

doi:10.3934/era.2022175

Electronic Research Archive

2022, Volume 30, Issue 9: 3435-3451. doi: 10.3934/era.2022175

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Research article

Zavadskij modules over cluster-tilted algebras of type $ \mathbb{A} $

1.
Departamento de Matemáticas, Universidad Nacional de Colombia, Edificio Yu Takeuchi 404, Kra 30 No 45-03, Bogotá 11001000, Colombia
2.
Escuela de Matemáticas y Estadística, Universidad Pedagógica y Tecnológica de Colombia, Avenida Central del Norte 39-115, Tunja 150003, Colombia

Academic Editor: Xiao-Wu Chen
^*In memory of Professor Daniel Simson (1942-2022).

Received: 23 May 2022 Revised: 14 July 2022 Accepted: 17 July 2022 Published: 21 July 2022

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.
- algorithm of differentiation,
- categorification,
- cluster-tilted algebra of type $ \mathbb{A} $,
- integer sequence,
- OEIS,
- quiver representation,
- uniserial module,
- Zavadskij module
Citation: Agustín Moreno Cañadas, Robinson-Julian Serna, Isaías David Marín Gaviria. Zavadskij modules over cluster-tilted algebras of type $ \mathbb{A} $[J]. Electronic Research Archive, 2022, 30(9): 3435-3451. doi: 10.3934/era.2022175

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Abstract

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.

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