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
A module pair （C, T） over an Artin algebra A is called a tilting pair if both C and T are selforthogonal modules and the conditions T e ada C and C ∈ add T hold. The duality on a tilting pair is investigated to dis...A module pair （C, T） over an Artin algebra A is called a tilting pair if both C and T are selforthogonal modules and the conditions T e ada C and C ∈ add T hold. The duality on a tilting pair is investigated to discuss the condition under which the dual of a tilting pair is also a tilting pair. A necessary and sufficient condition of （D（7）, D（C） ） being an n-tilting pair over an Artin algebra for an n-tilting pair （ C, 7） is given. And, a necessary and sufficient condition of （ T^＊, C^＊ ） being an ntilting pair over a selfinjective Artin algebra for an n-tilting pair （C, 7） is also given.展开更多
The authors give a discription of the finite representation type over an algebraically stable categories of selfinjective algebras of closed field, which admits indecomposable Calabi-Yau obdjects. For selfinjective al...The authors give a discription of the finite representation type over an algebraically stable categories of selfinjective algebras of closed field, which admits indecomposable Calabi-Yau obdjects. For selfinjective algebras with such properties, the ones whose stable categories are not Calabi-Yau are determined. For the remaining ones, i.e., those selfinjective algebras whose stable categories are actually Calabi-Yau, the difference between the Calabi-Yau dimensions of the indecomposable Calabi-Yau objects and the Calabi-Yau dimensions of the stable categories is described.展开更多
Let A be a finite-dimensional algebra over an algebraically closed field k,ε the category of all exact sequences in A-rood, Mp （respectively, Ml） the full subcategory of C consisting of those objects with projecti...Let A be a finite-dimensional algebra over an algebraically closed field k,ε the category of all exact sequences in A-rood, Mp （respectively, Ml） the full subcategory of C consisting of those objects with projective （respectively, injective） middle terms. It is proved that Mp （respectively, MI） is contravariantly finite （respectively, covariantly finite） in ε. As an application, it is shown that Mp = MI is functorially finite and has Auslander-Reiten sequences provided A is selfinjective. Keywords category of exact sequences, contravariantly finite subcategory, functorially finite subcategory Auslander-Reiten sequences, selfinjective algebra展开更多
Let G be a finite group and A be a finite-dimensional selfinjective algebra over an algebraically closed field. Suppose A is a left G-module algebra. Some suficient conditions for the skew group algebra AG to be stabl...Let G be a finite group and A be a finite-dimensional selfinjective algebra over an algebraically closed field. Suppose A is a left G-module algebra. Some suficient conditions for the skew group algebra AG to be stably Calabi-Yau are provided, and some new examples of stably Calabi-Yau algebras are given as well.展开更多
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.
基金The National Natural Science Foundation of China (No.10971024)the Specialized Research Fund for the Doctoral Program of Higher Education ( No. 200802860024)+1 种基金the Natural Science Foundation of Jiangsu Province ( No. BK2010393 )Scientific Research Foundation of Guangxi University ( No. XJZ100246)
文摘A module pair （C, T） over an Artin algebra A is called a tilting pair if both C and T are selforthogonal modules and the conditions T e ada C and C ∈ add T hold. The duality on a tilting pair is investigated to discuss the condition under which the dual of a tilting pair is also a tilting pair. A necessary and sufficient condition of （D（7）, D（C） ） being an n-tilting pair over an Artin algebra for an n-tilting pair （ C, 7） is given. And, a necessary and sufficient condition of （ T^＊, C^＊ ） being an ntilting pair over a selfinjective Artin algebra for an n-tilting pair （C, 7） is also given.
基金supported by the National Natural Science Foundation of China (No. 10801099)the Zhejiang Provincial Natural Science Foundation of China (No. J20080154)the grant from Science Technology Department of Zhejiang Province (No. 2011R10051)
文摘The authors give a discription of the finite representation type over an algebraically stable categories of selfinjective algebras of closed field, which admits indecomposable Calabi-Yau obdjects. For selfinjective algebras with such properties, the ones whose stable categories are not Calabi-Yau are determined. For the remaining ones, i.e., those selfinjective algebras whose stable categories are actually Calabi-Yau, the difference between the Calabi-Yau dimensions of the indecomposable Calabi-Yau objects and the Calabi-Yau dimensions of the stable categories is described.
基金supported by National Natural Science Foundation of China(Grant No.11271257)National Science Foundation of Shanghai Municiple(Granted No.13ZR1422500)
文摘Let A be a finite-dimensional algebra over an algebraically closed field k,ε the category of all exact sequences in A-rood, Mp （respectively, Ml） the full subcategory of C consisting of those objects with projective （respectively, injective） middle terms. It is proved that Mp （respectively, MI） is contravariantly finite （respectively, covariantly finite） in ε. As an application, it is shown that Mp = MI is functorially finite and has Auslander-Reiten sequences provided A is selfinjective. Keywords category of exact sequences, contravariantly finite subcategory, functorially finite subcategory Auslander-Reiten sequences, selfinjective algebra
基金supported by National Natural Science Foundation of China (GrantNo. 10971188)
文摘Let G be a finite group and A be a finite-dimensional selfinjective algebra over an algebraically closed field. Suppose A is a left G-module algebra. Some suficient conditions for the skew group algebra AG to be stably Calabi-Yau are provided, and some new examples of stably Calabi-Yau algebras are given as well.
