Let A be a basic connected finite dimensional associative algebra over an algebraically closed field k and G be a cyclic group.There is a quiver QGwith relationsρG such that the skew group algebras A[G]is Morita equi...Let A be a basic connected finite dimensional associative algebra over an algebraically closed field k and G be a cyclic group.There is a quiver QGwith relationsρG such that the skew group algebras A[G]is Morita equivalent to the quotient algebra of path algebra kQGmodulo ideal(ρG).Generally,the quiver QGis not connected.In this paper we develop a method to determine the number of connect components of QG.Meanwhile,we introduce the notion of weight for underlying quiver of A such that A is G-graded and determine the connect components of smash product A#kG*.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
Motivated by T-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a finite-dimensional algebra A with action by a finite group G, we introduce the notion of G-stable support τ-tilting modules. Then we es...Motivated by T-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a finite-dimensional algebra A with action by a finite group G, we introduce the notion of G-stable support τ-tilting modules. Then we establish bijections among G-stable support τ-tilting modules over ∧, G-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective ∧-modules, and G-stable functorially finite torsion classes in the category of finitely generated left ∧-modules. In the case when ∧ is the endomorphism of a G-stable cluster-tilting object T over a Horn-finite 2-Calabi- Yau triangulated category L with a G-action, these are also in bijection with G-stable cluster-tilting objects in L. Moreover, we investigate the relationship between stable support τ-tilitng modules over ∧ and the skew group algebra ∧G.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.11871404,11971398 and 12131018)。
文摘Let A be a basic connected finite dimensional associative algebra over an algebraically closed field k and G be a cyclic group.There is a quiver QGwith relationsρG such that the skew group algebras A[G]is Morita equivalent to the quotient algebra of path algebra kQGmodulo ideal(ρG).Generally,the quiver QGis not connected.In this paper we develop a method to determine the number of connect components of QG.Meanwhile,we introduce the notion of weight for underlying quiver of A such that A is G-graded and determine the connect components of smash product A#kG*.
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 authors would like to thank Dong Yang and Yuefei Zheng for their helpful discussion. This work was partially supported by the National Natural Science Foundation of China （Grant No. 11571164） and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘Motivated by T-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a finite-dimensional algebra A with action by a finite group G, we introduce the notion of G-stable support τ-tilting modules. Then we establish bijections among G-stable support τ-tilting modules over ∧, G-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective ∧-modules, and G-stable functorially finite torsion classes in the category of finitely generated left ∧-modules. In the case when ∧ is the endomorphism of a G-stable cluster-tilting object T over a Horn-finite 2-Calabi- Yau triangulated category L with a G-action, these are also in bijection with G-stable cluster-tilting objects in L. Moreover, we investigate the relationship between stable support τ-tilitng modules over ∧ and the skew group algebra ∧G.
