Let (K, M,H) be an upper triangular bimodule problem. Briistle and Hille showed that the opposite algebra A of the endomorphism algebra of a projective generator P of the matrices category of (K., M, H) is quasi-hered...Let (K, M,H) be an upper triangular bimodule problem. Briistle and Hille showed that the opposite algebra A of the endomorphism algebra of a projective generator P of the matrices category of (K., M, H) is quasi-hereditary, and there is an equivalence between the category of△-good modules of A and Mat(K, M). In this note, based on the tame theorem for bimodule problems, we show that if the algebra A associated with an upper triangular bimodule problem is of△-tame representation type, then the category F(△) has the homogeneous property, i.e. almost all modules in F(△) are isomorphic to their Auslander-Reiten translations. Moreover, if (K, M,H)is an upper triangular bipartite bimodule problem, then A is of△-tame representation type if and only if F(△) is homogeneous.展开更多
The bocs corresponding to each hereditary algebra of representation tame type is shown. And the minimal bocs obtained from it is given with only irreducible maps for each dimension d. The minimal bocs (after deleting ...The bocs corresponding to each hereditary algebra of representation tame type is shown. And the minimal bocs obtained from it is given with only irreducible maps for each dimension d. The minimal bocs (after deleting finitely many vertices) coincides with a full subquiver of the AR-quiver of hereditary algebra, in which the indecomposable modules M have the property dim (top M)<d.展开更多
In this paper, we study selfinjective Koszul algebras of finite complexity. We prove that the complexity is a nonnegative integer when it is finite; and that the category Yt of modules with complexity less or equal to...In this paper, we study selfinjective Koszul algebras of finite complexity. We prove that the complexity is a nonnegative integer when it is finite; and that the category Yt of modules with complexity less or equal to t, is resolving and coresolving. We show that for each 0 ≤ 1 ≤ m there exist a family of modules of complexity 1 parameterized by G（l, m）, the Grassmannian of l-dimensional subspaces of an m-dimensional vector space V, for the exterior algebra of V. Using complexity, we also give a new approach to the representation theory of a tame symmetric algebra with vanishing radical cube over an algebraically closed field of characteristic 0, via skew group algebra of a finite subgroup of SL（2, C） over the exterior algebra of a 2-dimensional vector space.展开更多
IN the theory of infinite abelian groups, the concepts of torsion-free groups and divisible groups are well known. We know that there exists, up to isomorphism, one and only one abelian group which is both torsion-fre...IN the theory of infinite abelian groups, the concepts of torsion-free groups and divisible groups are well known. We know that there exists, up to isomorphism, one and only one abelian group which is both torsion-free and divisible at the same time. This is the additive group Q of the rational numbers. Q plays an important role in the study of infinite-abelian groups. The theory of infinite-dirnensional modules over a finite-dimensional algebra runs similarly to the theory of infinite abelian groups but there are also some substantial differences.展开更多
Let A be a finite-dimensional algebra over an algebraically closed field. An indecomposable (right) ,4-module M is called generic provided M is infinite k-dimensional but finite length as (left) EndA(M)-module. Let R ...Let A be a finite-dimensional algebra over an algebraically closed field. An indecomposable (right) ,4-module M is called generic provided M is infinite k-dimensional but finite length as (left) EndA(M)-module. Let R = A DA be the trivial extension algebra of A- Generic R-modules are constructed from generic A-modules using some functors between Mod A and Mod R. it is also proved that if A is a tame hereditary algebra, then R has only two generic modules.展开更多
Let be a connected finite quiver without oriented cycle, A=k（?） the corresponding path algebra with k being an algebraically closed field, A<sup>T</sup> a preprojective tilting module. B=End<sub>A&...Let be a connected finite quiver without oriented cycle, A=k（?） the corresponding path algebra with k being an algebraically closed field, A<sup>T</sup> a preprojective tilting module. B=End<sub>A</sub>T. Then B is called a tame （resp. wild）concealed algebra provided is an Euclidean （resp. wild ） graph. The following result is important in the representation theory of tame concealed algebras （see [1,4.9]）: if A is tame concealed, T= T<sub>0</sub>⊕ T<sub>1</sub> a tilting module with T<sub>0</sub> nonzero preprojective and T<sub>1</sub> regular, then End<sub>A</sub>T<sub>0</sub> is tame concealed. The main purpose of this note is to generalize it to the 'wild' case. For this we generally consider the endomorphism algebra of preprojective partial tilting modules over a concealed algebra. For the notations the readers can refer to Ref.[1].展开更多
We give a complete classification of tilting bundles over a weighted projective line of type （2, 3, 3）. This yields another realization of the tame concealed algebras of type E6.
Ring epimorphisms often induce silting modules and cosilting modules,termed minimal silting or minimal cosilting.The aim of this paper is twofold.Firstly,we determine the minimal tilting and minimal cotilting modules ...Ring epimorphisms often induce silting modules and cosilting modules,termed minimal silting or minimal cosilting.The aim of this paper is twofold.Firstly,we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra.In particular,we show that a large cotilting module is minimal if and only if it has an adic module as a direct summand.Secondly,we discuss the behavior of minimality under ring extensions.We show that minimal cosilting modules over a commutative noetherian ring extend to minimal cosilting modules along any flat ring epimorphism.Similar results are obtained for commutative rings of small homological dimensions.展开更多
基金This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10426014,10501010 and 19331030)the Foundation of Hubei Provincial Department of Education (Grant No.D200510005).
文摘Let (K, M,H) be an upper triangular bimodule problem. Briistle and Hille showed that the opposite algebra A of the endomorphism algebra of a projective generator P of the matrices category of (K., M, H) is quasi-hereditary, and there is an equivalence between the category of△-good modules of A and Mat(K, M). In this note, based on the tame theorem for bimodule problems, we show that if the algebra A associated with an upper triangular bimodule problem is of△-tame representation type, then the category F(△) has the homogeneous property, i.e. almost all modules in F(△) are isomorphic to their Auslander-Reiten translations. Moreover, if (K, M,H)is an upper triangular bipartite bimodule problem, then A is of△-tame representation type if and only if F(△) is homogeneous.
文摘The bocs corresponding to each hereditary algebra of representation tame type is shown. And the minimal bocs obtained from it is given with only irreducible maps for each dimension d. The minimal bocs (after deleting finitely many vertices) coincides with a full subquiver of the AR-quiver of hereditary algebra, in which the indecomposable modules M have the property dim (top M)<d.
基金Supported by NSFC #10671061SRFDP #200505042004the Cultivation Fund of the Key Scientific and Technical Innovation Project #21000115 of the Ministry of Education of China
文摘In this paper, we study selfinjective Koszul algebras of finite complexity. We prove that the complexity is a nonnegative integer when it is finite; and that the category Yt of modules with complexity less or equal to t, is resolving and coresolving. We show that for each 0 ≤ 1 ≤ m there exist a family of modules of complexity 1 parameterized by G（l, m）, the Grassmannian of l-dimensional subspaces of an m-dimensional vector space V, for the exterior algebra of V. Using complexity, we also give a new approach to the representation theory of a tame symmetric algebra with vanishing radical cube over an algebraically closed field of characteristic 0, via skew group algebra of a finite subgroup of SL（2, C） over the exterior algebra of a 2-dimensional vector space.
文摘IN the theory of infinite abelian groups, the concepts of torsion-free groups and divisible groups are well known. We know that there exists, up to isomorphism, one and only one abelian group which is both torsion-free and divisible at the same time. This is the additive group Q of the rational numbers. Q plays an important role in the study of infinite-abelian groups. The theory of infinite-dirnensional modules over a finite-dimensional algebra runs similarly to the theory of infinite abelian groups but there are also some substantial differences.
文摘Let A be a finite-dimensional algebra over an algebraically closed field. An indecomposable (right) ,4-module M is called generic provided M is infinite k-dimensional but finite length as (left) EndA(M)-module. Let R = A DA be the trivial extension algebra of A- Generic R-modules are constructed from generic A-modules using some functors between Mod A and Mod R. it is also proved that if A is a tame hereditary algebra, then R has only two generic modules.
基金Project supported by the National Natural Science Foundation of China
文摘Let be a connected finite quiver without oriented cycle, A=k（?） the corresponding path algebra with k being an algebraically closed field, A<sup>T</sup> a preprojective tilting module. B=End<sub>A</sub>T. Then B is called a tame （resp. wild）concealed algebra provided is an Euclidean （resp. wild ） graph. The following result is important in the representation theory of tame concealed algebras （see [1,4.9]）: if A is tame concealed, T= T<sub>0</sub>⊕ T<sub>1</sub> a tilting module with T<sub>0</sub> nonzero preprojective and T<sub>1</sub> regular, then End<sub>A</sub>T<sub>0</sub> is tame concealed. The main purpose of this note is to generalize it to the 'wild' case. For this we generally consider the endomorphism algebra of preprojective partial tilting modules over a concealed algebra. For the notations the readers can refer to Ref.[1].
文摘We give a complete classification of tilting bundles over a weighted projective line of type （2, 3, 3）. This yields another realization of the tame concealed algebras of type E6.
基金supported by Fondazione Cariverona,Program“Ricerca Scientifica di Eccellenza 2018”(Project“Reducing Complexity in Algebra,Logic,Combinatorics-REDCOM”)supported by China Scholarship Council(Grant No.201906860022)。
文摘Ring epimorphisms often induce silting modules and cosilting modules,termed minimal silting or minimal cosilting.The aim of this paper is twofold.Firstly,we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra.In particular,we show that a large cotilting module is minimal if and only if it has an adic module as a direct summand.Secondly,we discuss the behavior of minimality under ring extensions.We show that minimal cosilting modules over a commutative noetherian ring extend to minimal cosilting modules along any flat ring epimorphism.Similar results are obtained for commutative rings of small homological dimensions.
