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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Bibliographic Details
Published inJournal of mathematics (Hidawi) Vol. 2023
Main Authors Hu, Desheng, Obul, Abdukadir
Format Journal Article
LanguageEnglish
Published Cairo Hindawi 2023
Hindawi Limited
Subjects
Algebra
Commutators
Mathematics
Polynomials
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Summary: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.
ISSN:2314-4629
2314-4785
DOI:10.1155/2023/6820173