The characteristic tilting modules of quasi-hereditary algebras which are dual extensions of directed monomial algebras are explicitly constructed; and it is shown that the Ringel dual of the dual extension of an arbi...The characteristic tilting modules of quasi-hereditary algebras which are dual extensions of directed monomial algebras are explicitly constructed; and it is shown that the Ringel dual of the dual extension of an arbitrary hereditary algebra has triangular decomposition and bipartite quiver.展开更多
Associated with each finite directed quiver Q is a quasi-hereditary algebra. the so-called twisted double of the path algebra kQ. Characteristic tilting modules over this class of quasi-hereditary algebras are constru...Associated with each finite directed quiver Q is a quasi-hereditary algebra. the so-called twisted double of the path algebra kQ. Characteristic tilting modules over this class of quasi-hereditary algebras are constructed. Their endonorphism algebras are explicitly described. It turns out that this class of quasi-hereditary algebras is closed under taking the Ringel dual.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
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
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
Let t be a positive integer and A be a hereditary abelian category satisfying some finiteness conditions.We define the semi-derived Ringel-Hall algebra of A from the category C_(Z/t)(A)of Z/t-graded complexes and obta...Let t be a positive integer and A be a hereditary abelian category satisfying some finiteness conditions.We define the semi-derived Ringel-Hall algebra of A from the category C_(Z/t)(A)of Z/t-graded complexes and obtain a natural basis of the semi-derived Ringel-Hall algebra.Moreover,we describe the semiderived Ringel-Hall algebra by the generators and defining relations.In particular,if t is an odd integer,we show an embedding of the derived Hall algebra of the odd-periodic relative derived category in the extended semi-derived Ringel-Hall algebra.展开更多
By using the Ringel-Hall algebra approach, we investigate the structure of the Lie algebra L（A） generated by indecomposable constructible sets in the varieties of modules for any finite- dimensional C-algebra A. We ...By using the Ringel-Hall algebra approach, we investigate the structure of the Lie algebra L（A） generated by indecomposable constructible sets in the varieties of modules for any finite- dimensional C-algebra A. We obtain a geometric realization of the universal enveloping algebra R（A） of L（A）, this generalizes the main result of Riedtmann. We also obtain Green＇s formula in a geometric form for any finite-dimensional C-algebra A and use it to give the comultiplication formula in R（A）.展开更多
In this paper, we construct the GrSbner-Shirshov bases for degenerate Ringel- Hall algebras of types A and G2 from the multiplication formulas of the corresponding generic extension monoid algebras.
In the present paper we describe a specialization of prinjective Ringel-Hall algebra to 1, for prinjective modules over incidence algebras of posets of finite prinjective type, by generators and relations. This gives ...In the present paper we describe a specialization of prinjective Ringel-Hall algebra to 1, for prinjective modules over incidence algebras of posets of finite prinjective type, by generators and relations. This gives us a generalisation of Serre relations for semisimple Lie algebras. Connections of prinjective Ringel-Hall algebras with classical Lie algebras are also discussed.展开更多
We use the Ringel-Hall algebra approach to study the canonical basis elements for the quantum group of type B2 which are characterized in Xi [12]. However, our approach simplifies several computations there.
For a given hereditary abelian category satisfying some finiteness conditions,in certain twisted cases it is shown that the modified Ringel-Hall algebra is isomorphic to the naive lattice algebra and there exists an e...For a given hereditary abelian category satisfying some finiteness conditions,in certain twisted cases it is shown that the modified Ringel-Hall algebra is isomorphic to the naive lattice algebra and there exists an epimorphism from the modified Ringel-Hall algebra to the lattice algebra.Furthermore,the kernel of this epimorphism is described explicitly.Finally,we show that the naive lattice algebra is invariant under the derived equivalences of hereditary abelian categories.展开更多
This paper is devoted to the study of the structure of the double Ringel-Hall algebra D（A） for an infinite dimensional hereditary algebra A, which is given by a valued quiver F over a finite field, and also to the ...This paper is devoted to the study of the structure of the double Ringel-Hall algebra D（A） for an infinite dimensional hereditary algebra A, which is given by a valued quiver F over a finite field, and also to the study of the relations of D（A）-modules with representations of valued quiver Г.展开更多
Abstract Let n ≥ 3. The complex Lie algebra, which is attached to a unit form xixj and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type A~, and r...Abstract Let n ≥ 3. The complex Lie algebra, which is attached to a unit form xixj and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type A~, and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.展开更多
文摘The characteristic tilting modules of quasi-hereditary algebras which are dual extensions of directed monomial algebras are explicitly constructed; and it is shown that the Ringel dual of the dual extension of an arbitrary hereditary algebra has triangular decomposition and bipartite quiver.
基金Supported partially by Alexander von Humboldt FoundationNational Natural Science Foundation of China (Grant No. 19971009).
文摘Associated with each finite directed quiver Q is a quasi-hereditary algebra. the so-called twisted double of the path algebra kQ. Characteristic tilting modules over this class of quasi-hereditary algebras are constructed. Their endonorphism algebras are explicitly described. It turns out that this class of quasi-hereditary algebras is closed under taking the Ringel dual.
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 Natural Science Foundation of China(Grant Nos.12001107 and 11821001)University Natural Science Project of Anhui Province(Grant No.KJ2021A0661)+1 种基金University Outstanding Youth Research Project in Anhui Province(Grant No.2022AH020082)Scientific Research and Innovation Team Project of Fuyang Normal University(Grant No.TDJC2021009)。
文摘Let t be a positive integer and A be a hereditary abelian category satisfying some finiteness conditions.We define the semi-derived Ringel-Hall algebra of A from the category C_(Z/t)(A)of Z/t-graded complexes and obtain a natural basis of the semi-derived Ringel-Hall algebra.Moreover,we describe the semiderived Ringel-Hall algebra by the generators and defining relations.In particular,if t is an odd integer,we show an embedding of the derived Hall algebra of the odd-periodic relative derived category in the extended semi-derived Ringel-Hall algebra.
基金Supported by NSF of China (Grant No. 10631010)by NKBRPC (Grant No. 2006CB805905)
文摘By using the Ringel-Hall algebra approach, we investigate the structure of the Lie algebra L（A） generated by indecomposable constructible sets in the varieties of modules for any finite- dimensional C-algebra A. We obtain a geometric realization of the universal enveloping algebra R（A） of L（A）, this generalizes the main result of Riedtmann. We also obtain Green＇s formula in a geometric form for any finite-dimensional C-algebra A and use it to give the comultiplication formula in R（A）.
文摘In this paper, we construct the GrSbner-Shirshov bases for degenerate Ringel- Hall algebras of types A and G2 from the multiplication formulas of the corresponding generic extension monoid algebras.
文摘In the present paper we describe a specialization of prinjective Ringel-Hall algebra to 1, for prinjective modules over incidence algebras of posets of finite prinjective type, by generators and relations. This gives us a generalisation of Serre relations for semisimple Lie algebras. Connections of prinjective Ringel-Hall algebras with classical Lie algebras are also discussed.
文摘We use the Ringel-Hall algebra approach to study the canonical basis elements for the quantum group of type B2 which are characterized in Xi [12]. However, our approach simplifies several computations there.
基金supported by National Natural Science Foundation of China(Grant No.11701473)Youth Talent Foundation of Fuyang Normal University(Grant No.rcxm201803)。
文摘For a given hereditary abelian category satisfying some finiteness conditions,in certain twisted cases it is shown that the modified Ringel-Hall algebra is isomorphic to the naive lattice algebra and there exists an epimorphism from the modified Ringel-Hall algebra to the lattice algebra.Furthermore,the kernel of this epimorphism is described explicitly.Finally,we show that the naive lattice algebra is invariant under the derived equivalences of hereditary abelian categories.
基金Project supported by the National Natural Science Foundation of China (No.10471071) the 973 Project of the Ministry of Science and Technology of China.
文摘This paper is devoted to the study of the structure of the double Ringel-Hall algebra D（A） for an infinite dimensional hereditary algebra A, which is given by a valued quiver F over a finite field, and also to the study of the relations of D（A）-modules with representations of valued quiver Г.
文摘Abstract Let n ≥ 3. The complex Lie algebra, which is attached to a unit form xixj and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type A~, and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.
