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
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
In this paper,we introduce Frobenius von Neumann algebras and study quantum convolution inequalities.In this framework,we unify quantum Young's inequality on quantum symmetries such as subfactors,and fusion bi-alg...In this paper,we introduce Frobenius von Neumann algebras and study quantum convolution inequalities.In this framework,we unify quantum Young's inequality on quantum symmetries such as subfactors,and fusion bi-algebras studied in quantum Fourier analysis.Moreover,we prove quantum entropic convolution inequalities and characterize the extremizers in the subfactor case.We also prove quantum smooth entropic convolution inequalities.We obtain the positivity of comultiplications of subfactor planar algebras,which is stronger than the quantum Schur product theorem.All these inequalities provide analytic obstructions of unitary categorification of fusion rings stronger than the Schur product criterion.展开更多
Khovanov type homology is a generalization of Khovanov homology.The main result of this paper is to give a recursive formula for Khovanov type homology of pretzel knots P（-n,-m, m）. The computations reveal that the ...Khovanov type homology is a generalization of Khovanov homology.The main result of this paper is to give a recursive formula for Khovanov type homology of pretzel knots P（-n,-m, m）. The computations reveal that the rank of the homology of pretzel knots is an invariant of n. The proof is based on a ＂shortcut＂ and two lemmas that recursively reduce the computational complexity of Khovanov type homology.展开更多
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
Let k be a fixed algebraically closed field of arbitrary characteristic,let Λ be a finite dimensional self-injective k-algebra,and let ∨ be an indecomposable non-projective left Λ-module with finite dimension over ...Let k be a fixed algebraically closed field of arbitrary characteristic,let Λ be a finite dimensional self-injective k-algebra,and let ∨ be an indecomposable non-projective left Λ-module with finite dimension over k.We prove that if τΛ∨ is the Auslander-Reiten translation of ∨,then the versal deformation rings R(Λ,∨)and R(Λ,τΛ∨)(in the sense of F.M.Bleher and the second author)are isomorphic.We use this to prove that if Λ is further a cluster-tilted k-algebra,then R(Λ,∨)is universal and isomorphic to k.展开更多
In this paper, by using the Frobenius morphism and the multiplication formulas of the generic extension monoid algebra, the authors first give a presentation of the degenerate Ringel-Hall algebra, and then construct t...In this paper, by using the Frobenius morphism and the multiplication formulas of the generic extension monoid algebra, the authors first give a presentation of the degenerate Ringel-Hall algebra, and then construct the Gr¨obner-Shirshov basis for degenerate Ringel-Hall algebras of type F_4.展开更多
In this paper, for rings R, we introduce complex rings C（R）, quaternion rings H（R）, and octonion rings O（R）, which are extension rings of R; R C（R） H（R） O（R）. Our main purpose of this paper is to sho...In this paper, for rings R, we introduce complex rings C（R）, quaternion rings H（R）, and octonion rings O（R）, which are extension rings of R; R C（R） H（R） O（R）. Our main purpose of this paper is to show that if R is a Frobenius algebra, then these extension rings are Frobenius Mgebras and if R is a quasi-Frobenius ring, then C（R） and H（R） are quasi-Frobenius rings and, when Char（R）=2, O（R） is also a quasi-Frobenius ring.展开更多
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.
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supported by Beijing Municipal Science&Technology Commission,Administrative Commission of Zhongguancun Science Park(Grant No.Z221100002722017)supported by grants from Beijing Institute of Mathematical Sciences and Applications+4 种基金supported by China’s National Key R&D Programmes(Grant No.2020YFA0713000)Tsinghua University(Grant No.04200100122)Templeton Religion Trust(TRT 159)supported by Beijing Natural Science Foundation Key Program(Grant No.Z220002)supported by National Natural Science Foundation of China(Grant Nos.12031004 and 12371124)。
文摘In this paper,we introduce Frobenius von Neumann algebras and study quantum convolution inequalities.In this framework,we unify quantum Young's inequality on quantum symmetries such as subfactors,and fusion bi-algebras studied in quantum Fourier analysis.Moreover,we prove quantum entropic convolution inequalities and characterize the extremizers in the subfactor case.We also prove quantum smooth entropic convolution inequalities.We obtain the positivity of comultiplications of subfactor planar algebras,which is stronger than the quantum Schur product theorem.All these inequalities provide analytic obstructions of unitary categorification of fusion rings stronger than the Schur product criterion.
基金The NSF(11271282,11371013)of Chinathe Graduate Innovation Fund of USTS
文摘Khovanov type homology is a generalization of Khovanov homology.The main result of this paper is to give a recursive formula for Khovanov type homology of pretzel knots P（-n,-m, m）. The computations reveal that the rank of the homology of pretzel knots is an invariant of n. The proof is based on a ＂shortcut＂ and two lemmas that recursively reduce the computational complexity of Khovanov type homology.
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supported by the Release Time for Research Scholarship of the Office of Academic Affairs and by the Faculty Research Seed Grant funded by the Office of Sponsored ProgramsResearch Administration at the Valdosta State University as well as partly supported by CODI and Estrategia de Sostenibilidad(Universidad de Antioquia,UdeA).
文摘Let k be a fixed algebraically closed field of arbitrary characteristic,let Λ be a finite dimensional self-injective k-algebra,and let ∨ be an indecomposable non-projective left Λ-module with finite dimension over k.We prove that if τΛ∨ is the Auslander-Reiten translation of ∨,then the versal deformation rings R(Λ,∨)and R(Λ,τΛ∨)(in the sense of F.M.Bleher and the second author)are isomorphic.We use this to prove that if Λ is further a cluster-tilted k-algebra,then R(Λ,∨)is universal and isomorphic to k.
基金Acknowledgements Part of the work was done during the author＇s visit to SCMS （Shanghai Center for Mathematical Sciences）, and the author would like to thank for the hospitality. The author also thank the referees for their careful reading, helpful suggestions and comments. This work was supported by the National Natural Science Foundation of China （Grant No. 11301180）.
文摘We compute explicitly the modular derivations for Poisson-Ore extensions and tensor products of Poisson algebras.
基金supported by the National Natural Science Foundation of China(Nos.11361056,11271043,11471186)
文摘In this paper, by using the Frobenius morphism and the multiplication formulas of the generic extension monoid algebra, the authors first give a presentation of the degenerate Ringel-Hall algebra, and then construct the Gr¨obner-Shirshov basis for degenerate Ringel-Hall algebras of type F_4.
文摘In this paper, for rings R, we introduce complex rings C（R）, quaternion rings H（R）, and octonion rings O（R）, which are extension rings of R; R C（R） H（R） O（R）. Our main purpose of this paper is to show that if R is a Frobenius algebra, then these extension rings are Frobenius Mgebras and if R is a quasi-Frobenius ring, then C（R） and H（R） are quasi-Frobenius rings and, when Char（R）=2, O（R） is also a quasi-Frobenius ring.
