We introduce the quiver of a bicomodule over a cosemisimple coalgebra. Applying this to the coradical C0 of an arbitrary coalgebra C, we give an alternative definition of the Gabriel quiver of C, and then show that it...We introduce the quiver of a bicomodule over a cosemisimple coalgebra. Applying this to the coradical C0 of an arbitrary coalgebra C, we give an alternative definition of the Gabriel quiver of C, and then show that it coincides with the known Ext quiver of C and the link quiver of C. The dual Gabriel theorem for a coalgebra with a separable coradical is obtained, which generalizes the corresponding result for a pointed coalgebra. We also give a new description of C1 = C0 ∧C C0 of any coalgebra C, which can be regarded as a generalization of the first part of the well-known Taft-Wilson Theorem for pointed coal-gebras. As applications, we give a characterization of locally finite coalgebras via their Gabriel quivers, and a property of the Gabriel quiver of a quasi-coFrobenius coalgebra.展开更多
Using the quiver technique we construct a class of non-graded bi-Frobenius algebras. We also classify a class of graded bi-Frobenius algebras via certain equations of structure coefficients.
The relationships between piecewise-Koszul algebras and other ＂Koszul-type＂ algebras are discussed. The Yoneda-Ext algebra and the dual algebra of a piecewise-Koszul algebra are studied, and a sufficient condition f...The relationships between piecewise-Koszul algebras and other ＂Koszul-type＂ algebras are discussed. The Yoneda-Ext algebra and the dual algebra of a piecewise-Koszul algebra are studied, and a sufficient condition for the dual algebra A^1 to be piecewise-Koszul is given. Finally, by studying the trivial extension algebras of the path algebras of Dynkin quivers in bipartite orientation, we give explicit constructions for piecewise-Koszul algebras with arbitrary ＂period＂ and piecewise-Koszul algebras with arbitrary ＂jump-degree＂.展开更多
We single out a class of translation quivers and prove combinatorially that the translation quivers in this class are coils. These coils form a class of special coils. They are easier to visualize, but still show all ...We single out a class of translation quivers and prove combinatorially that the translation quivers in this class are coils. These coils form a class of special coils. They are easier to visualize, but still show all the strange behaviour of general coils, and contain quasi-stable tubes as special examples.展开更多
For each finite subgroup G of SLn（C）, we introduce the generalized Cartan matrix AG in view of McKay correspondence from the fusion rule of its natural representation. Using group theory, we show that the generalize...For each finite subgroup G of SLn（C）, we introduce the generalized Cartan matrix AG in view of McKay correspondence from the fusion rule of its natural representation. Using group theory, we show that the generalized Cartan matrices have similar favorable properties such as positive semi- definiteness as in the classical case of affine Cartan matrices. The complete McKay quivers for SL3 （C） are explicitly described and classified based on representation theory.展开更多
We single out a class of translation quivers and prove that the translation quivers in this class are generalized multicoils. These generalized multicoils form a class of special generalized multicoils. They are easie...We single out a class of translation quivers and prove that the translation quivers in this class are generalized multicoils. These generalized multicoils form a class of special generalized multicoils. They are easier to visualize, but still show all the strange behaviours of generalized multicoils, and contain quasi-stable tubes as special examples.展开更多
The representations of the Dynkin quivers and the corresponding Euclidean quivers are treated in many books. These notes provide three building blocks for dealing with representations of Dynkin （and Euclidean） quive...The representations of the Dynkin quivers and the corresponding Euclidean quivers are treated in many books. These notes provide three building blocks for dealing with representations of Dynkin （and Euclidean） quivers. They should be helpful as part of a direct approach to study representations of quivers, and they shed some new light on properties of Dynkin and Euclidean quivers.展开更多
Let K be an algebraically closed field and A be a finite dimensional algebra over K. In this paper we give a classification of biserial incidence algebras with quiver methods.
The goal of this paper is to investigate whether the Ext-groups of all pairs (M, N) of modules over Nakayama algebras of type (n,n,n) satisfy the condition ExtΛn(M,N)=0 for n >> 0 ? ExtΛn(M,N)=0 for n >>...The goal of this paper is to investigate whether the Ext-groups of all pairs (M, N) of modules over Nakayama algebras of type (n,n,n) satisfy the condition ExtΛn(M,N)=0 for n >> 0 ? ExtΛn(M,N)=0 for n >> 0. We achieve that by discussing the Ext-groups of Nakayama algebra with projectives of lengths 3n+1 and 3n+2 using combinations of modules of different lengths.展开更多
Let F be an algebraically closed field, A be a finite dimensional algebra over F, Г A be the Auslander Reiten quiver of A, Γ be a connected component of Γ A with oriented cycles and semi stable vertices and each no...Let F be an algebraically closed field, A be a finite dimensional algebra over F, Г A be the Auslander Reiten quiver of A, Γ be a connected component of Γ A with oriented cycles and semi stable vertices and each non stable vertex in Γ be a projective injective vertex. The structure of Γ is studied.展开更多
We prove that the transition matrix between a special Poincaré-Birkhoff-Witt(PBW)basis and the semicanonical basis of U+(sln(C))is upper triangular and unipotent under any order which is compatible with the parti...We prove that the transition matrix between a special Poincaré-Birkhoff-Witt(PBW)basis and the semicanonical basis of U+(sln(C))is upper triangular and unipotent under any order which is compatible with the partial order deg.展开更多
We study properties of graded maximal Cohen-Macaulay modules over an N-graded locally finite,Auslander Gorenstein,and Cohen-Macaulay algebra of dimension two.As a consequence,we extend a part of the McKay corresponden...We study properties of graded maximal Cohen-Macaulay modules over an N-graded locally finite,Auslander Gorenstein,and Cohen-Macaulay algebra of dimension two.As a consequence,we extend a part of the McKay correspondence in dimension two to a more general setting.展开更多
In this paper, we prove that there exists no sectional cycle in a translation quiver under certain conditions. So, we generalize Dautista and Smalφ's corresponding result on AR-quiver of an artin algebra.
基金supported by the National Natural Science Foundation of China(Grant Nos.10271113&10301033)the Doctoral Foundation of the Chinese Education Ministry.
文摘We introduce the quiver of a bicomodule over a cosemisimple coalgebra. Applying this to the coradical C0 of an arbitrary coalgebra C, we give an alternative definition of the Gabriel quiver of C, and then show that it coincides with the known Ext quiver of C and the link quiver of C. The dual Gabriel theorem for a coalgebra with a separable coradical is obtained, which generalizes the corresponding result for a pointed coalgebra. We also give a new description of C1 = C0 ∧C C0 of any coalgebra C, which can be regarded as a generalization of the first part of the well-known Taft-Wilson Theorem for pointed coal-gebras. As applications, we give a characterization of locally finite coalgebras via their Gabriel quivers, and a property of the Gabriel quiver of a quasi-coFrobenius coalgebra.
基金This work was supported in part by the National Natural Science Foundation of China(Grant Nos.10501041,10271113,10601052)
文摘Using the quiver technique we construct a class of non-graded bi-Frobenius algebras. We also classify a class of graded bi-Frobenius algebras via certain equations of structure coefficients.
基金Supported by National Natural Science Foundation of China （Grant No. 11001245）, Zhejiang Province Department of Education Fund （Grant No. Y201016432） and Zhejiang Innovation Project （Grant No. T200905）
文摘The relationships between piecewise-Koszul algebras and other ＂Koszul-type＂ algebras are discussed. The Yoneda-Ext algebra and the dual algebra of a piecewise-Koszul algebra are studied, and a sufficient condition for the dual algebra A^1 to be piecewise-Koszul is given. Finally, by studying the trivial extension algebras of the path algebras of Dynkin quivers in bipartite orientation, we give explicit constructions for piecewise-Koszul algebras with arbitrary ＂period＂ and piecewise-Koszul algebras with arbitrary ＂jump-degree＂.
文摘We single out a class of translation quivers and prove combinatorially that the translation quivers in this class are coils. These coils form a class of special coils. They are easier to visualize, but still show all the strange behaviour of general coils, and contain quasi-stable tubes as special examples.
基金supported by National Natural Science Foundation of China (Grant No. 10728102)National Security Agency (Grant No. MDA 904-97-1-0062)
文摘For each finite subgroup G of SLn（C）, we introduce the generalized Cartan matrix AG in view of McKay correspondence from the fusion rule of its natural representation. Using group theory, we show that the generalized Cartan matrices have similar favorable properties such as positive semi- definiteness as in the classical case of affine Cartan matrices. The complete McKay quivers for SL3 （C） are explicitly described and classified based on representation theory.
文摘We single out a class of translation quivers and prove that the translation quivers in this class are generalized multicoils. These generalized multicoils form a class of special generalized multicoils. They are easier to visualize, but still show all the strange behaviours of generalized multicoils, and contain quasi-stable tubes as special examples.
文摘The representations of the Dynkin quivers and the corresponding Euclidean quivers are treated in many books. These notes provide three building blocks for dealing with representations of Dynkin （and Euclidean） quivers. They should be helpful as part of a direct approach to study representations of quivers, and they shed some new light on properties of Dynkin and Euclidean quivers.
基金Foundation item： Supported by the National Natural Science Foundation of China（11271119） Supported by the Natural Science Foundation of Beijing（1122002）
文摘Let K be an algebraically closed field and A be a finite dimensional algebra over K. In this paper we give a classification of biserial incidence algebras with quiver methods.
文摘The goal of this paper is to investigate whether the Ext-groups of all pairs (M, N) of modules over Nakayama algebras of type (n,n,n) satisfy the condition ExtΛn(M,N)=0 for n >> 0 ? ExtΛn(M,N)=0 for n >> 0. We achieve that by discussing the Ext-groups of Nakayama algebra with projectives of lengths 3n+1 and 3n+2 using combinations of modules of different lengths.
文摘Let F be an algebraically closed field, A be a finite dimensional algebra over F, Г A be the Auslander Reiten quiver of A, Γ be a connected component of Γ A with oriented cycles and semi stable vertices and each non stable vertex in Γ be a projective injective vertex. The structure of Γ is studied.
基金supported by National Natural Science Foundation of China (Grant Nos.11171183 and 11371165)Program for Changjiang Scholars and Innovative Research Team in University (Grant No.IRT1264)
文摘We prove that the transition matrix between a special Poincaré-Birkhoff-Witt(PBW)basis and the semicanonical basis of U+(sln(C))is upper triangular and unipotent under any order which is compatible with the partial order deg.
基金The authors thank the referees for the careful reading and very useful suggestions and thank Ken Brown,Daniel Rogalski,Robert Won,and Quanshui Wu for many useful conversations and valuable comments on the subject.Y.-H.Wang and X.-S.Qin thank the Department of Mathematics,University of Washington for its very supportive hospitality during their visitsX.-S.Qin was partially supported by the Foundation of China Scholarship Council(Grant No.[2016]3100)+3 种基金Y.-H.Wang was partially supported by the National Natural Science Foundation of China(Grant Nos.11971289,11871071)the Foundation of Shanghai Science and Technology Committee(Grant No.15511107300)the Foundation of China Scholarship Council(Grant No.[2016]3009)J.J.Zhang was partially supported by the US National Science Foundation(Grant No.DMS-1700825).
文摘We study properties of graded maximal Cohen-Macaulay modules over an N-graded locally finite,Auslander Gorenstein,and Cohen-Macaulay algebra of dimension two.As a consequence,we extend a part of the McKay correspondence in dimension two to a more general setting.
基金Supported by Fund of Educational Ministry of China and the Fund of Education Committee of Beijing.
文摘In this paper, we prove that there exists no sectional cycle in a translation quiver under certain conditions. So, we generalize Dautista and Smalφ's corresponding result on AR-quiver of an artin algebra.
