Non-isomorphic two dimensional indecomposable modules over infinite dimensional hereditary path algebras are described. We infer that none of them can be determined by their dimension vectors.
Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed feld of characteristic zero.We describe all annihilator ideals of indecomposable H-modules by generators.In particular,we giv...Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed feld of characteristic zero.We describe all annihilator ideals of indecomposable H-modules by generators.In particular,we give the classification of all ideals of the finite-dimensional pointed rank one Hopf algebra of nilpotent type over the Klein 4-group.展开更多
Let R be a commutative Noetherian ring and p be a prime ideal of R such that the ideal pRp is principal and ht(p)≠0. In this note, the anthors describe the explicit structure of the injective envelope of the R-module...Let R be a commutative Noetherian ring and p be a prime ideal of R such that the ideal pRp is principal and ht(p)≠0. In this note, the anthors describe the explicit structure of the injective envelope of the R-module R/p.展开更多
Let A be a subalgebra of uq（sl（2）） generated by K, K-1 and F and A^δ be a subalgebra of b/q（sl（2）） generated by K, K-1 （and also Fd if q is a primitive d-th root of unity with d an odd number）. Given an Aδ...Let A be a subalgebra of uq（sl（2）） generated by K, K-1 and F and A^δ be a subalgebra of b/q（sl（2）） generated by K, K-1 （and also Fd if q is a primitive d-th root of unity with d an odd number）. Given an Aδ-module M, a/gq（s/（2））-module A A M is constructed via the iterated Ore extension of Uq（S/（2）） in a unified framework for any q. Then all the submodules of A δA5 M are determined for a fixed finite-dimensional indecomposable Aδ-module .M. It turns out that for some indecomposable A^-module M, the 5/q（sl（2））-module A @A M is indecomposable, which is not in the BGG-categories associated with quantum groups in general.展开更多
In this paper, by using the Ringel-Hall algebra method, we prove that the set of the skew-commutator relations of quantum root vectors forms a minimal GrSbner- Shirshov basis for the quantum groups of Dynkin type. As ...In this paper, by using the Ringel-Hall algebra method, we prove that the set of the skew-commutator relations of quantum root vectors forms a minimal GrSbner- Shirshov basis for the quantum groups of Dynkin type. As an application, we give an explicit basis for the types E7 and Dn.展开更多
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 ...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 set of these skew-commutators is a GrSbner-Shirshov basis for quantum group of type D4.展开更多
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...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 minimal GrSbner-Shirshov basis for the twisted generic composition algebras of affine type.展开更多
In this paper,we give a Grobner-Shirshov basis of quantum group of type C3 by using the Ringel-Hall algebra approach.For this,first we compute all skew-commutator relations between the isoclasses of indecomposable rep...In this paper,we give a Grobner-Shirshov basis of quantum group of type C3 by using the Ringel-Hall algebra approach.For 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”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;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 byproduct.Then we prove that the set of these relations is closed under composition.So they constitutes a minimal Grobner-Shirshov basis of the positive part of quantum group of type C3.Dually,we get a Grobner-Shirshov basis of the negative part of quantum group of type C3.And finally we give a Grobner-Shirshov basis for the whole quantum group of type C3.展开更多
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...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.展开更多
Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero.In this paper we show that any finitedimensional indecomposable H-module is generated by one ele...Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero.In this paper we show that any finitedimensional indecomposable H-module is generated by one element.In particular,any indecomposable submodule of H under the adjoint action is generated by a special element of H.Using this result,we show that the Hopf algebra H is a principal ideal ring,i.e.,any two-sided ideal of H is generated by one element.As an application,we give explicitly the generators of ideals,primitive ideals,maximal ideals and completely prime ideals of the Taft algebras.展开更多
Let G be a finite p-solvable group and k be an algebraic closed field of characteristic p. It is proved that any projective indecomposable module of a G-graded k-algebra is an induced module of a module of the subalge...Let G be a finite p-solvable group and k be an algebraic closed field of characteristic p. It is proved that any projective indecomposable module of a G-graded k-algebra is an induced module of a module of the subalgebra graded by a Hall p′-subgroup. A necessary and sufficient condition for the indecomposability of an induced module from a Hall p′-subgroup is obtained.展开更多
基金The NSF(11371307)of ChinaResearch Culture Funds(2014xmpy11)of Anhui Normal University
文摘Non-isomorphic two dimensional indecomposable modules over infinite dimensional hereditary path algebras are described. We infer that none of them can be determined by their dimension vectors.
基金supported by the National Natural Science Foundation of China(Grant No.12371041).
文摘Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed feld of characteristic zero.We describe all annihilator ideals of indecomposable H-modules by generators.In particular,we give the classification of all ideals of the finite-dimensional pointed rank one Hopf algebra of nilpotent type over the Klein 4-group.
基金This research is in part supported by a grant from IPM.
文摘Let R be a commutative Noetherian ring and p be a prime ideal of R such that the ideal pRp is principal and ht(p)≠0. In this note, the anthors describe the explicit structure of the injective envelope of the R-module R/p.
基金Supported by National Natural Foundation of China (Grant No. 11171291)Doctorate Foundation (Grant No. 200811170001) Ministry of Education of China
文摘Let A be a subalgebra of uq（sl（2）） generated by K, K-1 and F and A^δ be a subalgebra of b/q（sl（2）） generated by K, K-1 （and also Fd if q is a primitive d-th root of unity with d an odd number）. Given an Aδ-module M, a/gq（s/（2））-module A A M is constructed via the iterated Ore extension of Uq（S/（2）） in a unified framework for any q. Then all the submodules of A δA5 M are determined for a fixed finite-dimensional indecomposable Aδ-module .M. It turns out that for some indecomposable A^-module M, the 5/q（sl（2））-module A @A M is indecomposable, which is not in the BGG-categories associated with quantum groups in general.
基金Supported by the National Natural Science Foundation of China （11061033）.Acknowledgements. Part of this work is done during the corresponding author＇s visiting the Stuttgart University with the support of China Scholarship Council. With this opportunity, he expresses his gratefulness to Professor Steffen Koenig and the Institute of Algebra and Number Theory of Stuttgart University and the China Scholarship Council.
文摘In this paper, by using the Ringel-Hall algebra method, we prove that the set of the skew-commutator relations of quantum root vectors forms a minimal GrSbner- Shirshov basis for the quantum groups of Dynkin type. As an application, we give an explicit basis for the types E7 and Dn.
基金Project supported by the Natural Science Foundation of Xinjiang University (the Starting Research Fund for Doctors) (No. BS080103)
文摘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 set of these skew-commutators is a GrSbner-Shirshov basis for quantum group of type D4.
基金Supported by National Science Foundation of China （Grant No. 11061033, 11361056）.
文摘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 minimal GrSbner-Shirshov basis for the twisted generic composition algebras of affine type.
基金This paper is supported by the National Natural Science Foundation of China(No.11061033).
文摘In this paper,we give a Grobner-Shirshov basis of quantum group of type C3 by using the Ringel-Hall algebra approach.For 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”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;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 byproduct.Then we prove that the set of these relations is closed under composition.So they constitutes a minimal Grobner-Shirshov basis of the positive part of quantum group of type C3.Dually,we get a Grobner-Shirshov basis of the negative part of quantum group of type C3.And finally we give a Grobner-Shirshov basis for the whole quantum group of type C3.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant No. ]1061033）.
文摘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.
基金supported by the National Natural Science Foundation of China(Grant No.11871063)supported by the Qing Lan project.
文摘Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero.In this paper we show that any finitedimensional indecomposable H-module is generated by one element.In particular,any indecomposable submodule of H under the adjoint action is generated by a special element of H.Using this result,we show that the Hopf algebra H is a principal ideal ring,i.e.,any two-sided ideal of H is generated by one element.As an application,we give explicitly the generators of ideals,primitive ideals,maximal ideals and completely prime ideals of the Taft algebras.
文摘Let G be a finite p-solvable group and k be an algebraic closed field of characteristic p. It is proved that any projective indecomposable module of a G-graded k-algebra is an induced module of a module of the subalgebra graded by a Hall p′-subgroup. A necessary and sufficient condition for the indecomposability of an induced module from a Hall p′-subgroup is obtained.
