Invariant properties of modules under smash products from finite dimensional algebras
We give the relationship between indecomposable modules over the finite dimensional $ k $-algebra $ A $ and the smash product $ A\sharp G $ respectively, where $ G $ is a finite abelian group satisfying $ G\subseteq Aut(A) $, and $ k $ is an algebraically closed field with the characteristic not div...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-01-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023342?viewType=HTML |
| _version_ | 1811176162379956224 |
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| author | Wanwan Jia Fang Li |
| author_facet | Wanwan Jia Fang Li |
| author_sort | Wanwan Jia |
| collection | DOAJ |
| description | We give the relationship between indecomposable modules over the finite dimensional $ k $-algebra $ A $ and the smash product $ A\sharp G $ respectively, where $ G $ is a finite abelian group satisfying $ G\subseteq Aut(A) $, and $ k $ is an algebraically closed field with the characteristic not dividing the order of $ G $. More precisely, we construct all indecomposable $ A\sharp G $-modules from indecomposable $ A $-modules and prove that an $ A\sharp G $-module is indecomposable if and only if it is an indecomposable $ G $-stable module over $ A $. Besides, we give the relationship between simple, projective and injective modules in $ modA $ and those in $ modA\sharp G $. |
| first_indexed | 2024-04-10T19:47:45Z |
| format | Article |
| id | doaj.art-c2910ff1c43c4f999043a5061eb5c7ee |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-10T19:47:45Z |
| publishDate | 2023-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-c2910ff1c43c4f999043a5061eb5c7ee2023-01-29T02:31:18ZengAIMS PressAIMS Mathematics2473-69882023-01-01836737674810.3934/math.2023342Invariant properties of modules under smash products from finite dimensional algebrasWanwan Jia0Fang Li1Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, ChinaDepartment of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, ChinaWe give the relationship between indecomposable modules over the finite dimensional $ k $-algebra $ A $ and the smash product $ A\sharp G $ respectively, where $ G $ is a finite abelian group satisfying $ G\subseteq Aut(A) $, and $ k $ is an algebraically closed field with the characteristic not dividing the order of $ G $. More precisely, we construct all indecomposable $ A\sharp G $-modules from indecomposable $ A $-modules and prove that an $ A\sharp G $-module is indecomposable if and only if it is an indecomposable $ G $-stable module over $ A $. Besides, we give the relationship between simple, projective and injective modules in $ modA $ and those in $ modA\sharp G $. https://www.aimspress.com/article/doi/10.3934/math.2023342?viewType=HTMLfinite dimensional algebrasmash productindecomposable moduleg-stable modulestable categoryabelian |
| spellingShingle | Wanwan Jia Fang Li Invariant properties of modules under smash products from finite dimensional algebras AIMS Mathematics finite dimensional algebra smash product indecomposable module g-stable module stable category abelian |
| title | Invariant properties of modules under smash products from finite dimensional algebras |
| title_full | Invariant properties of modules under smash products from finite dimensional algebras |
| title_fullStr | Invariant properties of modules under smash products from finite dimensional algebras |
| title_full_unstemmed | Invariant properties of modules under smash products from finite dimensional algebras |
| title_short | Invariant properties of modules under smash products from finite dimensional algebras |
| title_sort | invariant properties of modules under smash products from finite dimensional algebras |
| topic | finite dimensional algebra smash product indecomposable module g-stable module stable category abelian |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023342?viewType=HTML |
| work_keys_str_mv | AT wanwanjia invariantpropertiesofmodulesundersmashproductsfromfinitedimensionalalgebras AT fangli invariantpropertiesofmodulesundersmashproductsfromfinitedimensionalalgebras |