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 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
This paper proves that every indecomposable module over a representation-finite selfinjective algebra of class An is uniquely determined by its Loewy factors and every indecomposable module is multiplicity-free. As an...This paper proves that every indecomposable module over a representation-finite selfinjective algebra of class An is uniquely determined by its Loewy factors and every indecomposable module is multiplicity-free. As an application to the representation theory of finite groups, it is obtained that every indecomposable module over blocks with cycle defect groups is uniquely determined by its Loewy factors and every indecomposable module is multiplicity-free.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
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.
<正> The q-deformation of Verma theory for the Lie algebra is studied in this paper. Theindecomposable representations and the induced representations of quantum universal envelop-ing algebra sl_q(3) are constru...<正> The q-deformation of Verma theory for the Lie algebra is studied in this paper. Theindecomposable representations and the induced representations of quantum universal envelop-ing algebra sl_q(3) are constructed on the q-deformed Verma space and the quotient spacesrespectively. We put stress on the discussion of the case in which q is a root of unity. Usingthe new representation constrained in the subalgebra sl_q(2), we systematically constructthe new series of solutions (Yang-Baxter matrices) for Yang-Baxter equation without spectralparameter.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 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
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.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.
文摘This paper proves that every indecomposable module over a representation-finite selfinjective algebra of class An is uniquely determined by its Loewy factors and every indecomposable module is multiplicity-free. As an application to the representation theory of finite groups, it is obtained that every indecomposable module over blocks with cycle defect groups is uniquely determined by its Loewy factors and every indecomposable module is multiplicity-free.
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Project supported in part by the National Natural Science Foundation of China.
文摘<正> The q-deformation of Verma theory for the Lie algebra is studied in this paper. Theindecomposable representations and the induced representations of quantum universal envelop-ing algebra sl_q(3) are constructed on the q-deformed Verma space and the quotient spacesrespectively. We put stress on the discussion of the case in which q is a root of unity. Usingthe new representation constrained in the subalgebra sl_q(2), we systematically constructthe new series of solutions (Yang-Baxter matrices) for Yang-Baxter equation without spectralparameter.
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.
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.
