A Clifford deformation of a Koszul Frobenius algebra E is a finite dimensional Z_(2)-graded algebra E(θ),which corresponds to a noncommutative quadric hypersurface E^(!)/(z)for some central regular element z∈E_(2)^(...A Clifford deformation of a Koszul Frobenius algebra E is a finite dimensional Z_(2)-graded algebra E(θ),which corresponds to a noncommutative quadric hypersurface E^(!)/(z)for some central regular element z∈E_(2)^(!).It turns out that the bounded derived category D^(b)(gr_(Z_(2))E(θ))is equivalent to the stable category of the maximal Cohen-Macaulay modules over E^(!)/(z)provided that E!is noetherian.As a consequence,E^(!)/(z)is a noncommutative isolated singularity if and only if the corresponding Clifford deformation E(θ)is a semisimple Z_(2)-graded algebra.The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius algebra are related to Knörrer's periodicity theorem for quadric hypersurfaces.As an application,we recover Knörrer's periodicity theorem without using matrix factorizations.展开更多
Some equivalent conditions for double Frobenius algebras to be strict ones are given. Then some examples of （strict or non-strict） double Frobenius algebras are presented. Finally, a sufficient and necessary conditi...Some equivalent conditions for double Frobenius algebras to be strict ones are given. Then some examples of （strict or non-strict） double Frobenius algebras are presented. Finally, a sufficient and necessary condition for the trivial extension of a double Frobenius algebra to be a （strict） double Frobenius algebra is given.展开更多
Let (H, R) be a co-Frobenius quasitriangular Hopf algebra with antipode S. Denote the set of group-like elements in H by G (H). In this paper, we find a necessary and sufficient condition for (H, R) to have a ribbon e...Let (H, R) be a co-Frobenius quasitriangular Hopf algebra with antipode S. Denote the set of group-like elements in H by G (H). In this paper, we find a necessary and sufficient condition for (H, R) to have a ribbon element. The condition gives a connection with the order of G (H) and the order of S2.展开更多
Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A^H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H...Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A^H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H to be Frobenius over A. Using the theory of A-rings, we mainly construct a Morita context 〈A^H,A#H,A,A,τ,μ〉 connecting the smash product A#H and the invariant subalgebra A^H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.展开更多
In this paper, we generalize two kinds of graded algebras, δ-Koszul algebras and Kp algebras, to the non-graded cases. The trivial modules of δ-Koszul algebras have pure resolutions, while those of Kp algebras admit...In this paper, we generalize two kinds of graded algebras, δ-Koszul algebras and Kp algebras, to the non-graded cases. The trivial modules of δ-Koszul algebras have pure resolutions, while those of Kp algebras admit non-pure resolutions. We provide necessary and sufficient conditions for a notherian semiperfect algebra either to be a quasi-δ-Koszul algebra or to be a quasi-Kp algebra.展开更多
Let A be a Frobenius k-algebra. The matrix algebra R =(■) is called a generalized matrix algebra over a Frobenius algebra A. In this paper we show that R is also a Frobenius algebra.
In this paper,we compute the Frobenius dimension of any cluster-tilted algebra of finite type.Moreover,we give conditions on the bound quiver of a cluster-tilted algebra A such that八has non-trivial open Frobenius str...In this paper,we compute the Frobenius dimension of any cluster-tilted algebra of finite type.Moreover,we give conditions on the bound quiver of a cluster-tilted algebra A such that八has non-trivial open Frobenius structures.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.
Let k be the algebraic closure of a finite field F_q and A be a finite dimensional k-algebra with a Frobenius morphism F.In the present paper we establish a relation between the stable module category of the repetitiv...Let k be the algebraic closure of a finite field F_q and A be a finite dimensional k-algebra with a Frobenius morphism F.In the present paper we establish a relation between the stable module category of the repetitive algebra of A and that of the repetitive algebra of the fixed-point algebra A^F.As an application,it is shown that the derived category of A^F is equivalent to the subcategory of F-stable objects in the derived category of A when A has a finite global dimension.展开更多
This paper is devoted to study Frobenius Poisson algebras. We introduce pseudo-unimodular Poisson algebras by generalizing unimodular Poisson algebras, and investigate Batalin-Vilkovisky structures on their cohomology...This paper is devoted to study Frobenius Poisson algebras. We introduce pseudo-unimodular Poisson algebras by generalizing unimodular Poisson algebras, and investigate Batalin-Vilkovisky structures on their cohomology algebras. For any Frobenius Poisson algebra, all Eatalin-Vilkovisky opera tors on its Poisson cochain complex are described explicitly. It is proved that there exists a Batalin-Vilkovisky operator on its cohomology algebra which is induced from a Batalin-Vilkovisky operator on the Poisson cochain complex, if and only if the Poisson st rue ture is pseudo-unimodular. The relation bet ween modular derivations of polynomial Poisson algebras and those of their truncated Poisson algebras is also described in some cases.展开更多
The aim of this paper is mainly to build a new representation-theoretic realization of finite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any ...The aim of this paper is mainly to build a new representation-theoretic realization of finite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any field. Finally, we give an analog of APR-tilting module for this class of algebras. The major conclusions contains the known results as special cases, e.g., that for path algebras over an algebraically closed field and for path algebras with relations from symmetrizable cartan matrices. Meanwhile, it means the corresponding results for some other important classes of algebras, that is, the path algebras of quivers over Frobenius algebras and the generalized path algebras endowed by Frobenius algebras at vertices.展开更多
In the present paper we describe a class of ep-Auslander-Yoneda algebras over K[χ]/（χ-n） in terms of quivers with relations, and prove that they are actually cellular algebras in the sense of Graham and Lehrer.
基金supported by ZJNSF(LY19A010011)NSFC(11971141,12371017)supported by NSFC(11971449,12131015,12371042).
文摘A Clifford deformation of a Koszul Frobenius algebra E is a finite dimensional Z_(2)-graded algebra E(θ),which corresponds to a noncommutative quadric hypersurface E^(!)/(z)for some central regular element z∈E_(2)^(!).It turns out that the bounded derived category D^(b)(gr_(Z_(2))E(θ))is equivalent to the stable category of the maximal Cohen-Macaulay modules over E^(!)/(z)provided that E!is noetherian.As a consequence,E^(!)/(z)is a noncommutative isolated singularity if and only if the corresponding Clifford deformation E(θ)is a semisimple Z_(2)-graded algebra.The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius algebra are related to Knörrer's periodicity theorem for quadric hypersurfaces.As an application,we recover Knörrer's periodicity theorem without using matrix factorizations.
基金This work was supported by the National Natural Science Foundation of China （Grant No. 11471282）, the China Postdoctoral Science Foundation （Grant No. 2017M610316）, and the Natural Science Foundation of Jiangsu Province （Grant No. BK20170589）.
文摘Some equivalent conditions for double Frobenius algebras to be strict ones are given. Then some examples of （strict or non-strict） double Frobenius algebras are presented. Finally, a sufficient and necessary condition for the trivial extension of a double Frobenius algebra to be a （strict） double Frobenius algebra is given.
文摘Let (H, R) be a co-Frobenius quasitriangular Hopf algebra with antipode S. Denote the set of group-like elements in H by G (H). In this paper, we find a necessary and sufficient condition for (H, R) to have a ribbon element. The condition gives a connection with the order of G (H) and the order of S2.
基金Supported by the NSF of China(1097104910971052)+1 种基金the NSF of Hebei Province(A2008000135A2009000253)
文摘Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A^H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H to be Frobenius over A. Using the theory of A-rings, we mainly construct a Morita context 〈A^H,A#H,A,A,τ,μ〉 connecting the smash product A#H and the invariant subalgebra A^H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.
基金Supported by the National Natural Science Foundation of China(10971188)the Zhejiang ProvincialNatural Science Foundation of China(J20080154)
文摘In this paper, we generalize two kinds of graded algebras, δ-Koszul algebras and Kp algebras, to the non-graded cases. The trivial modules of δ-Koszul algebras have pure resolutions, while those of Kp algebras admit non-pure resolutions. We provide necessary and sufficient conditions for a notherian semiperfect algebra either to be a quasi-δ-Koszul algebra or to be a quasi-Kp algebra.
基金The NSF(KJ2016A545,1808085MA14,KJ2018A0839) of Anhui Province
文摘Let A be a Frobenius k-algebra. The matrix algebra R =(■) is called a generalized matrix algebra over a Frobenius algebra A. In this paper we show that R is also a Frobenius algebra.
文摘In this paper,we compute the Frobenius dimension of any cluster-tilted algebra of finite type.Moreover,we give conditions on the bound quiver of a cluster-tilted algebra A such that八has non-trivial open Frobenius structures.
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the National Natural Science Foundation of China (Grant No.10671016)the 985 Project of Beijing Normal University
文摘Let k be the algebraic closure of a finite field F_q and A be a finite dimensional k-algebra with a Frobenius morphism F.In the present paper we establish a relation between the stable module category of the repetitive algebra of A and that of the repetitive algebra of the fixed-point algebra A^F.As an application,it is shown that the derived category of A^F is equivalent to the subcategory of F-stable objects in the derived category of A when A has a finite global dimension.
基金the School Foundation of Shanghai Normal University (project SK201712)the National Natural Science Foundation of China (Grant Nos. 11301180, 11771085)the National Natural Science Foundation of China (Grant Nos. 11771085, 11331006).
文摘This paper is devoted to study Frobenius Poisson algebras. We introduce pseudo-unimodular Poisson algebras by generalizing unimodular Poisson algebras, and investigate Batalin-Vilkovisky structures on their cohomology algebras. For any Frobenius Poisson algebra, all Eatalin-Vilkovisky opera tors on its Poisson cochain complex are described explicitly. It is proved that there exists a Batalin-Vilkovisky operator on its cohomology algebra which is induced from a Batalin-Vilkovisky operator on the Poisson cochain complex, if and only if the Poisson st rue ture is pseudo-unimodular. The relation bet ween modular derivations of polynomial Poisson algebras and those of their truncated Poisson algebras is also described in some cases.
基金Supported by National Natural Science Foundation of China(Grant Nos.11271318 and 11571173)the Zhejiang Provincial Natural Science Foundation of China(Grant No.LZ13A010001)
文摘The aim of this paper is mainly to build a new representation-theoretic realization of finite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any field. Finally, we give an analog of APR-tilting module for this class of algebras. The major conclusions contains the known results as special cases, e.g., that for path algebras over an algebraically closed field and for path algebras with relations from symmetrizable cartan matrices. Meanwhile, it means the corresponding results for some other important classes of algebras, that is, the path algebras of quivers over Frobenius algebras and the generalized path algebras endowed by Frobenius algebras at vertices.
文摘In the present paper we describe a class of ep-Auslander-Yoneda algebras over K[χ]/（χ-n） in terms of quivers with relations, and prove that they are actually cellular algebras in the sense of Graham and Lehrer.
