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PBW-Basis of Reduced Drinfeld Double Hall Algebra of Type An via Grobner-Shirshov Basis

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Algebra Colloquium 2023年第1期30卷 43-60页

作者： Yao Luo abdukadir obul College of Mathematics and System Sciences Xinjiang University Urumqi 830046China

In the Ringel-Hall algebra of Dynkin type,the set of all commutator relations between the isoclasses of indecomposable representations forms a minimal Grobner-Shirshov basis and the set of the corresponding irreducible elements forms a PBW-type basis of the Ringel-Hall *** aim to generalize this result to the reduced Drinfeld double Hall algebra of type A_(n).First,we compute a minimal Grobner-Shirshov basis for the reduced Drinfeld double Hall algebra of type An by proving that all possible compositions between the commutator relations are ***,by taking the corresponding irreducible monomials,we construct a PBW-type basis for the reduced Drinfeld double Hall algebra of type A_(n).

关键词： Drinfeld double Hall algebra indecomposable representations commutator relations Grobner-Shirshov basis

Grbner-Shirshov Basis of Quantum Group of Type D_4

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Chinese Annals of Mathematics,Series B 2011年第4期32卷 581-592页

作者： Gulshadam YUNUS abdukadir obul College of Mathematics and System Sciences Xinjiang University

The authors take all isomorphism classes of indecomposable representations as new generators, and obtain all skew-commutators between these generators by using the Ringel-Hall algebra method. Then they prove that the ... 详细信息

关键词： Ringel-Hall algebra Indecomposable modules GrSbner-Shirshov basis Compositions

Counting the number of indecomposable representations and a q-analogue of the Weyl-Kac denominator identity of type _n

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Science China Mathematics 2008年第6期51卷 1027-1035页

作者： obul abdukadir College of Mathematics and System Sciences Xinjiang UniversityUrumqi 830046China

By using Frobenius maps and F-stable representations,we count the number of isomor- phism classes of indecomposable representations with the fixed dimension vector of a species of type _n over a finite field,first,a... 详细信息

关键词： Frobenius map F-stable representation quiver with automorphism denominator identity

Grbner-Shirshov Bases of Irreducible Modules of the Quantum Group of Type G_2

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Chinese Annals of Mathematics,Series B 2016年第3期37卷 427-440页

作者： Ghani USTA abdukadir obul College of Mathematics and System Sciences Xinjiang University Urumqi 830046 China

First, the authors give a GrSbner-Shirshov basis of the finite-dimensional irre- ducible module Vq（λ） of the Drinfeld-Jimbo quantum group Uq（G2） by using the double free module method and the known GrSbner-Shirshov basis of Uq（G2）. Then, by specializing a suitable version of Uq （G2） at q = 1, they get a GrSbner-Shirshov basis of the universal enveloping algebra U（G2） of the simple Lie algebra of type G2 and the finite-dimensional irreducible U（G2）-module V（λ）.

关键词： Quantum group GrSbner-Shirshov basis Double free module Indecom-posable module Highest weight module

Grobner-Shirshov Basis of Twisted Generic Composition Algebras of Affine Type

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Algebra Colloquium 2017年第1期24卷 109-122页

作者： Gulshadam Yunus abdukadir obul College of Mathematics and System Sciences Xinjiang University Urumqi 830046 China

In this paper, by using PBW bases for the twisted generic composition algebras of affine type, we prove that the set of the skew-commutator relations of the iso-classes of indecomposable representations forms a minima... 详细信息

关键词： Ringel-Hall algebra indecomposable modules Grobner-Shirshov basis skew-commutator relations

The Skew-commutator Relations and Grobner-Shirshov Bases of Quantum Group of Type c3

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Acta Mathematicae Applicatae Sinica 2020年第4期36卷 825-835页

作者： Cheng-xiu QIANG abdukadir obul College of Electrical and Information Engineering Lanzhou University of TechnologyLanzhou 730050China College of Mathematics and System Science Xinjiang UniversityUrumqi 830046China

In this paper,we give a Grobner-Shirshov basis of quantum group of type C3 by using the Ringel-Hall algebra *** this,first we compute all skew-commutator relations between the isoclasses of indecomposable reprersentations of Ringel-Hall algebras of type C3 by using an“inductive”***,we do not use the traditional way of computing the skew-commutative relations,that is first compute all Hall polynomials then compute the corresponding skew-commutator relations;contrarily,we compute the“easier”skew-commutator relations which corresponding to those exact sequences with middile term indecomposable or the split exact sequences first,then“inductive”others from these“easier”ones and this in turn gives Hall polynomials as a *** we prove that the set of these relations is closed under *** they constitutes a minimal Grobner-Shirshov basis of the positive part of quantum group of type ***,we get a Grobner-Shirshov basis of the negative part of quantum group of type *** finally we give a Grobner-Shirshov basis for the whole quantum group of type C3.

关键词： Ringel-Hall algebras root vectors indecomposable modules Grobner-Shirshov bases compositions

Skew-commutator relations and Grobner- Shirshov basis of quantum group of type F4

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Frontiers of Mathematics in China 2014年第1期9卷 135-150页

作者： Chengxiu QIANG abdukadir obul College of Mathematics and System Sciences Xinjiang University Urumqi 830046 China

We give a Grobner-Shirshov basis of quantum group of type F4 by using the Ringel-Hall algebra approach. We compute all skew-commutator relations between the isoclasses of indecomposable representations of Ringel- Hall algebras of type F4 by using an ＇inductive＇ method. Precisely, we do not use the traditional way of computing the skew-commutative relations, that is first compute all Hall polynomials then compute the corresponding skew- commutator relations; instead, we compute the ＇easier＇ skew-commutator relations which correspond to those exact sequences with middle term indecomposable or the split exact sequences first, then ＇deduce＇ others from these ＇easier＇ ones and this in turn gives Hall polynomials as a byproduct. Then using the composition-diamond lemma prove that the set of these relations constitute a minimal CrSbner-Shirshov basis of the positive part of the quantum group of type F4. Dually, we get a Grobner-Shirshov basis of the negative part of the quantum group of type F4. And finally, we give a Gr6bner-Shirshov basis for the whole quantum group of type F4.

关键词： Ringel-Hall algebra root vector indecomposable module Grobner-Shirshov basis

Grbner-Shirshov Basis for Degenerate Ringel-Hall Algebras of Type F_4

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Chinese Annals of Mathematics,Series B 2016年第2期37卷 199-210页

作者： Zhenzhen GAO abdukadir obul College of Applied Sciences Beijing University of Technology College of Mathematics and System Sciences Xinjiang University

In this paper, by using the Frobenius morphism and the multiplication formulas of the generic extension monoid algebra, the authors first give a presentation of the degenerate Ringel-Hall algebra, and then construct t... 详细信息

关键词： Grbner-Shirshov basis Frobenius map Degenerate Ringel-Hall algebras Multiplication formulas

Grbner-Shirshov Basis and the Minimal Projective Resolution of U_q^+(G_2)

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Acta Mathematica Sinica,English Series 2016年第2期32卷 213-226页

作者： Ling Ling MAO abdukadir obul College of Mathematics and System Sciences Xinjiang UniversityUrumqi 830046P.R.Chin

Abstract In this paper, by using the Anick＇s resolution and Grobner-Shirshov basis for quantized enveloping algebra of type G2, we compute the minimal projective resolution of the trivial module of Uq＋（G2） and as... 详细信息

关键词： Anick resolution minimal projective resolution Grobner-Shirshov basis