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Research Article | Open Access

The counting formula for indecomposable modules over string algebra

Haicun Wen1( ), Mian-Tao Liu2( ), Yu-Zhe Liu3
Lanzhou Resources & Environment Voc-Tech University, Lanzhou 730022, China
Department of Mathematics, Nanjing University, Nanjing 210093, China
School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China
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Abstract

Let A=kQ/I be a string algebra. We show that, if for any vertex v of its bound quiver (Q,I), there exists at most one arrow (resp. at most two arrows) ending with v and there exist at most two arrows (resp. at most one arrow) starting with v, then the number of indecomposable modules over A is dimk⁡A+Σ, where Σ is induced by radP(v) (resp. E(v)/socE(v)) with decomposable socle (resp. top), where P(v) (resp. E(v)) is the indecomposable projective (resp. injective) module corresponded by the vertex v.

Keywords

representation-finiterepresentations of quiversindecomposable modules
CLC number: 16G10, 16G20, 16G60

References

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AIMS Mathematics
Volume 9 Issue 9,
September 2024
Pages 24977-24988
DOI: 10.3934/math.20241217

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Cite this article:
Wen H, Liu M-T, Liu Y-Z. The counting formula for indecomposable modules over string algebra. AIMS Mathematics, 2024, 9(9): 24977-24988. https://doi.org/10.3934/math.20241217

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Received: 19 June 2024
Revised: 09 August 2024
Accepted: 14 August 2024
Published: 15 September 2024
©2024 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)