PBW Basis of the Modified Ringel–Hall Algebra of Type An via Gröbner–Shirshov Basis
In the Ringel–Hall algebra of Dynkin type, the set of all commutator relations between the isoclasses of indecomposable representations forms a minimal Gröbner–Shirshov basis, and the set of the corresponding irreducible elements forms a PBW-type basis of the Ringel–Hall algebra. The aim of this pap...
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| Format: | Article |
| Language: | English |
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Wiley
2023-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2023/6820173 |
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| author | Desheng Hu Abdukadir Obul |
| author_facet | Desheng Hu Abdukadir Obul |
| author_sort | Desheng Hu |
| collection | DOAJ |
| description | In the Ringel–Hall algebra of Dynkin type, the set of all commutator relations between the isoclasses of indecomposable representations forms a minimal Gröbner–Shirshov basis, and the set of the corresponding irreducible elements forms a PBW-type basis of the Ringel–Hall algebra. The aim of this paper is to generalize this result to the modified Ringel–Hall algebra of type. First, we compute a minimal Gröbner–Shirshov basis for the modified Ringel–Hall algebra of type An by proving all possible compositions between the commutator relations are trivial. Then, by taking the corresponding irreducible monomials, we construct a PBW-type basis for the modified Ringel–Hall algebra of type An. |
| format | Article |
| id | doaj-art-9cc766c9e8f84a34b03cc8e3f068ca7e |
| institution | DOAJ |
| issn | 2314-4785 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-9cc766c9e8f84a34b03cc8e3f068ca7e2025-08-20T03:19:50ZengWileyJournal of Mathematics2314-47852023-01-01202310.1155/2023/6820173PBW Basis of the Modified Ringel–Hall Algebra of Type An via Gröbner–Shirshov BasisDesheng Hu0Abdukadir Obul1College of Mathematics and System Sciences Xinjiang UniversityCollege of Mathematics and System Sciences Xinjiang UniversityIn the Ringel–Hall algebra of Dynkin type, the set of all commutator relations between the isoclasses of indecomposable representations forms a minimal Gröbner–Shirshov basis, and the set of the corresponding irreducible elements forms a PBW-type basis of the Ringel–Hall algebra. The aim of this paper is to generalize this result to the modified Ringel–Hall algebra of type. First, we compute a minimal Gröbner–Shirshov basis for the modified Ringel–Hall algebra of type An by proving all possible compositions between the commutator relations are trivial. Then, by taking the corresponding irreducible monomials, we construct a PBW-type basis for the modified Ringel–Hall algebra of type An.http://dx.doi.org/10.1155/2023/6820173 |
| spellingShingle | Desheng Hu Abdukadir Obul PBW Basis of the Modified Ringel–Hall Algebra of Type An via Gröbner–Shirshov Basis Journal of Mathematics |
| title | PBW Basis of the Modified Ringel–Hall Algebra of Type An via Gröbner–Shirshov Basis |
| title_full | PBW Basis of the Modified Ringel–Hall Algebra of Type An via Gröbner–Shirshov Basis |
| title_fullStr | PBW Basis of the Modified Ringel–Hall Algebra of Type An via Gröbner–Shirshov Basis |
| title_full_unstemmed | PBW Basis of the Modified Ringel–Hall Algebra of Type An via Gröbner–Shirshov Basis |
| title_short | PBW Basis of the Modified Ringel–Hall Algebra of Type An via Gröbner–Shirshov Basis |
| title_sort | pbw basis of the modified ringel hall algebra of type an via grobner shirshov basis |
| url | http://dx.doi.org/10.1155/2023/6820173 |
| work_keys_str_mv | AT deshenghu pbwbasisofthemodifiedringelhallalgebraoftypeanviagrobnershirshovbasis AT abdukadirobul pbwbasisofthemodifiedringelhallalgebraoftypeanviagrobnershirshovbasis |