Let X be a compact complex manifold. Consider a small deformation π ： X → B of X, the dimensions of the cohomology groups of tangent sheaf Hq（xt, Txt ） may vary under this deformation. This article studies such p...Let X be a compact complex manifold. Consider a small deformation π ： X → B of X, the dimensions of the cohomology groups of tangent sheaf Hq（xt, Txt ） may vary under this deformation. This article studies such phenomena by studying the obstructions to deform a class in Hq（X, 5TX） with parameter t and gets a formula for the obstructions.展开更多
Let N be a maximal discrete nest on an infinite-dimensional separable Hilbert space H,ξ=∑^(∞)_(n=1)en/2n be a separating vector for the commutant N',E_(ξ)be the projection from H onto the subspace[Cξ]spanned ...Let N be a maximal discrete nest on an infinite-dimensional separable Hilbert space H,ξ=∑^(∞)_(n=1)en/2n be a separating vector for the commutant N',E_(ξ)be the projection from H onto the subspace[Cξ]spanned by the vectorξ,and Q be the projection from K=H⊕H⊕H onto the closed subspace{(η,η,η)^(T):η∈H}.Suppose that L is the projection lattice generated by the projections(E_(ξ) 0 0 0 0 0 0 0 0),{(E 0 0 0 0 0 0 0 0):E∈N},(I 0 0 0 I 0 0 0 0) and Q.We show that L is a Kadison-Singer lattice with the trivial commutant.Moreover,we prove that every n-th bounded cohomology group H~n(AlgL,B(K))with coefficients in B(K)is trivial for n≥1.展开更多
In this paper, we reveal that a weak entwining structure admits a rich cohomology theory. As an application we compute the cohomology of a weak entwining structure associated to a weak coalgebra-Galois extension.
The authors prove that all n-th completely bounded cohomology groups of a nest algebra T（N） acting on a separable Hilbert space are trivial when the coefficients lie in any ultraweakly closed T（N）-bimodule contain...The authors prove that all n-th completely bounded cohomology groups of a nest algebra T（N） acting on a separable Hilbert space are trivial when the coefficients lie in any ultraweakly closed T（N）-bimodule containing the nest algebra. They also prove that Hcb^n（A, M） ≌ Hcb^n（A, M） for all n ≥ 1 and a CSL algebra .A with an ultraweakly closed .A-bimodul.M containing A.展开更多
For any prime p, all finite noncyclic p-groups which contain a self-centralizing cyclic normal subgroup are determined by using cohomological techniques. Some applications are given, including a character theoretic de...For any prime p, all finite noncyclic p-groups which contain a self-centralizing cyclic normal subgroup are determined by using cohomological techniques. Some applications are given, including a character theoretic description for such groups.展开更多
A mechanics system consisting of three mass points on sphere S 2 is considered. The configuration space of the system is a fibre bundle over S 2 . It is proved that first Chern class of the bundle is -2 c 1...A mechanics system consisting of three mass points on sphere S 2 is considered. The configuration space of the system is a fibre bundle over S 2 . It is proved that first Chern class of the bundle is -2 c 1(γ) where γ is the canonical line bundle over the complex projective space CP 1=S 2 , which shows the bundle is non trivial. The information about the first Chern class makes the cohomology groups and homotopy groups of the configuration space worked out. In addition the effects of these topolo gical properties of the configuration space on the behavior in large scale of the system, as the number of equilibrium positions, periodic orbits and reduced phase space, are discussed.展开更多
This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W（2, 2） algebra. We show that the q-deformed W（2, 2） algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspon...This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W（2, 2） algebra. We show that the q-deformed W（2, 2） algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W（2, 2） algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras.As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W（2, 2） algebra.展开更多
We explicitly compute the first and second cohomology groups of the SchrSdinger algebra S（1） with coefficients in the trivial module and the finite-dimensional irreducible modules. We also show that the first and se...We explicitly compute the first and second cohomology groups of the SchrSdinger algebra S（1） with coefficients in the trivial module and the finite-dimensional irreducible modules. We also show that the first and second cohomology groups of S（1） with coefficients in the universal enveloping algebras U（S（1）） （under the adjoint action） are infinite dimensional.展开更多
We compute the derivations of the positive part of the two-parameter quantum group Ur,s（B3） and show that the Hochschild cohomology group of degree 1 of this algebra is a three- dimensional vector space over the bas...We compute the derivations of the positive part of the two-parameter quantum group Ur,s（B3） and show that the Hochschild cohomology group of degree 1 of this algebra is a three- dimensional vector space over the base field C. We also compute the groups of （Hopf） algebra automorphisms of the augmented two-parameter quantized enveloping algebra Ur,s（B3）.展开更多
In this paper,we compute the derivations of the positive part of the two-parameter quantum group of type G_(2) by embedding it into a quantum torus.We also show that the first Hochschild cohomology group of this algeb...In this paper,we compute the derivations of the positive part of the two-parameter quantum group of type G_(2) by embedding it into a quantum torus.We also show that the first Hochschild cohomology group of this algebra is a two-dimensional vector space over the complex field.展开更多
In this paper, let A be a finite dimensional associative algebra over an algebraically closed field h, modA be the category of finite dimensional left A-module and X1,X2,... ,X2, in modA be a complete exceptional sequ...In this paper, let A be a finite dimensional associative algebra over an algebraically closed field h, modA be the category of finite dimensional left A-module and X1,X2,... ,X2, in modA be a complete exceptional sequence, and let E be the endomorphism algebra of X1, X2,..., Xn. We study the global dimension of E, and calculate the Hochschild cohomology and homology groups of E.展开更多
Let G be a finite group and k a field of characteristic p】0.In this paper we consider the support variety for the cohomology module Ext<sub>kG</sub><sup>*</sup>（M,N） where M and N are kG-mod...Let G be a finite group and k a field of characteristic p】0.In this paper we consider the support variety for the cohomology module Ext<sub>kG</sub><sup>*</sup>（M,N） where M and N are kG-modules.It is the subvariety of the maximal ideal spectrum of H<sup>*</sup>（G,k） of the annihilator of the cohomology module. For modules in the principal block we show that that the variety is contained in the intersections of the varieties of M and N and the difference between the that intersection and the support variety of the cohomology module is contained in the group theoretic nucleus.For other blocks a new nucleus is defined and a similar theorem is proven.However in the case of modules in a nonprincipal block several new difficulties are highlighted by some examples.展开更多
In this paper we mainly study the derivations for even part of the finite-dimensional odd Hamiltonian superalgebra HO over a field of prime characteristic. We first give the generating set of the even part g of HO. Th...In this paper we mainly study the derivations for even part of the finite-dimensional odd Hamiltonian superalgebra HO over a field of prime characteristic. We first give the generating set of the even part g of HO. Then we compute the derivations from g into the even part m of the generalized Witt superalgebra. Finally, we determine the derivation algebra and outer derivation algebra of and the dimension formulas. In particular, the first cohomology groups H^1（g;m） and H^1（g;g） are determined.展开更多
In the present note that grew out of my talk given at the conference in honor of Prof. Zhong Tongde,I give a survey of some recent results about holomorphic vector bundles over general Hopf manifolds.
The author introduces a notion of weakIsequences and characterizes such sequences by means of homological methods.This notion extends the notion of weakMsequences and thus extends the notions of generalized Cohen Maca...The author introduces a notion of weakIsequences and characterizes such sequences by means of homological methods.This notion extends the notion of weakMsequences and thus extends the notions of generalized Cohen Macaulay modules and Buchsbaum modules to more general cases.展开更多
Let A be a connected minimal representation-infinite algebra over an alge- braically closed field k. In this paper, we investigate the simple connectedness and strong simple connectedness of A. We prove that A is simp...Let A be a connected minimal representation-infinite algebra over an alge- braically closed field k. In this paper, we investigate the simple connectedness and strong simple connectedness of A. We prove that A is simply connected if and only if its first Hochschild cohomology group Hi （A） is trivial. We also give some equivalent conditions of strong simple connectedness of an algebra A.展开更多
In this paper we consider the isomorphic classes of H-Galois extensions with the normal basis property over a fixed commutative ring B for a finite Hopf algebra H.The result is that when H is cocommutative,then the ab...In this paper we consider the isomorphic classes of H-Galois extensions with the normal basis property over a fixed commutative ring B for a finite Hopf algebra H.The result is that when H is cocommutative,then the above isomorphic classes and the 2-nd cohomology group H^2(L~*,U) are isomorphic as groups under the product defined in this paper,where L=BH,L~*=Hom_B(L,B).展开更多
This paper concerns a certain subcategory of the category of representations for a semisimple algebraic group G in characteristic p, which arises from the semisimple modules for the corresponding quantum group at a p-...This paper concerns a certain subcategory of the category of representations for a semisimple algebraic group G in characteristic p, which arises from the semisimple modules for the corresponding quantum group at a p-th root of unity. The subcategory, thus, records the cohomological difference between quantum groups and algebraic groups. We define translation functors in this category and use them to obtain information on the irreducible characters for G when the Lusztig character formula does not hold.展开更多
For a commutative ring R with a unit, an R-homology rose is a topological space whose homology groups with R-coefficients agree with those of a bouquet of circles. In this paper, we study some special properties of th...For a commutative ring R with a unit, an R-homology rose is a topological space whose homology groups with R-coefficients agree with those of a bouquet of circles. In this paper, we study some special properties of the fundamental groups of R-homology roses and their covering spaces, from which we obtain some results supporting the Carlsson conjecture on free(Zp)ractions. In addition, we discuss how to search candidates of the counterexamples of Wall's D(2)-problem among R-homology roses and R-acyclic spaces and propose some candidates.展开更多
基金partially supported by China-France-Russian mathematics collaboration grant,No.34000-3275100,from Sun Yat-Sen University
文摘Let X be a compact complex manifold. Consider a small deformation π ： X → B of X, the dimensions of the cohomology groups of tangent sheaf Hq（xt, Txt ） may vary under this deformation. This article studies such phenomena by studying the obstructions to deform a class in Hq（X, 5TX） with parameter t and gets a formula for the obstructions.
基金supported by National Natural Science Foundation of China(Grant No.11801342)Natural Science Foundation of Shaanxi Province(Grant No.2023-JC-YB-043)Shaanxi College Students Innovation and Entrepreneurship Training Program(Grant No.S202110708069)。
文摘Let N be a maximal discrete nest on an infinite-dimensional separable Hilbert space H,ξ=∑^(∞)_(n=1)en/2n be a separating vector for the commutant N',E_(ξ)be the projection from H onto the subspace[Cξ]spanned by the vectorξ,and Q be the projection from K=H⊕H⊕H onto the closed subspace{(η,η,η)^(T):η∈H}.Suppose that L is the projection lattice generated by the projections(E_(ξ) 0 0 0 0 0 0 0 0),{(E 0 0 0 0 0 0 0 0):E∈N},(I 0 0 0 I 0 0 0 0) and Q.We show that L is a Kadison-Singer lattice with the trivial commutant.Moreover,we prove that every n-th bounded cohomology group H~n(AlgL,B(K))with coefficients in B(K)is trivial for n≥1.
文摘In this paper, we reveal that a weak entwining structure admits a rich cohomology theory. As an application we compute the cohomology of a weak entwining structure associated to a weak coalgebra-Galois extension.
基金Supported partially by NSF of China (10201007)National Tianyuan Foundation of China (A0324614)
文摘The authors prove that all n-th completely bounded cohomology groups of a nest algebra T（N） acting on a separable Hilbert space are trivial when the coefficients lie in any ultraweakly closed T（N）-bimodule containing the nest algebra. They also prove that Hcb^n（A, M） ≌ Hcb^n（A, M） for all n ≥ 1 and a CSL algebra .A with an ultraweakly closed .A-bimodul.M containing A.
基金Supported by the NSF of China(11171194)by the NSF of Shanxi Province(2012011001-1)
文摘For any prime p, all finite noncyclic p-groups which contain a self-centralizing cyclic normal subgroup are determined by using cohomological techniques. Some applications are given, including a character theoretic description for such groups.
文摘A mechanics system consisting of three mass points on sphere S 2 is considered. The configuration space of the system is a fibre bundle over S 2 . It is proved that first Chern class of the bundle is -2 c 1(γ) where γ is the canonical line bundle over the complex projective space CP 1=S 2 , which shows the bundle is non trivial. The information about the first Chern class makes the cohomology groups and homotopy groups of the configuration space worked out. In addition the effects of these topolo gical properties of the configuration space on the behavior in large scale of the system, as the number of equilibrium positions, periodic orbits and reduced phase space, are discussed.
基金Supported by China Scholarship Council(Grant No.201206125047)China Postdoctoral Science Foundation Funded Project(Grant No.2012M520715)the Fundamental Research Funds for the Central Universities(Grant No.HIT.NSRIF.201462)
文摘This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W（2, 2） algebra. We show that the q-deformed W（2, 2） algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W（2, 2） algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras.As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W（2, 2） algebra.
基金Supported by National Natural Science Foundation of China(Grant No.11271056)Jiangsu Overseas Research & Training Program for University Prominent Young & Middle-aged Teachers and Presidentsthe Fundamental Research Funds for the Central Universities
文摘We explicitly compute the first and second cohomology groups of the SchrSdinger algebra S（1） with coefficients in the trivial module and the finite-dimensional irreducible modules. We also show that the first and second cohomology groups of S（1） with coefficients in the universal enveloping algebras U（S（1）） （under the adjoint action） are infinite dimensional.
基金supported by Specialized Research Fund for the Doctoral Program of Highter Education(Grant No.20130031110005)supported by NSFC(Grant No.11271131)
文摘We compute the derivations of the positive part of the two-parameter quantum group Ur,s（B3） and show that the Hochschild cohomology group of degree 1 of this algebra is a three- dimensional vector space over the base field C. We also compute the groups of （Hopf） algebra automorphisms of the augmented two-parameter quantized enveloping algebra Ur,s（B3）.
基金Supported by National Natural Science Foundation of China(Grant No.11771069)Natural Science Foundation of Heilongjiang Province(Grant No.LH2020A020)。
文摘In this paper,we compute the derivations of the positive part of the two-parameter quantum group of type G_(2) by embedding it into a quantum torus.We also show that the first Hochschild cohomology group of this algebra is a two-dimensional vector space over the complex field.
基金the National Natural Science Foundation of China （No. 10371036 10671061）+2 种基金 the Natural Science Foundation of Beijing （1042001） the Fund of Beijing Education Committee （No. KM200610005024） Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality and the Fundamental reseaxch Fund of Beijing University of Technology.
文摘In this paper, let A be a finite dimensional associative algebra over an algebraically closed field h, modA be the category of finite dimensional left A-module and X1,X2,... ,X2, in modA be a complete exceptional sequence, and let E be the endomorphism algebra of X1, X2,..., Xn. We study the global dimension of E, and calculate the Hochschild cohomology and homology groups of E.
基金Partially supported by grants from NSF and EPSRC
文摘Let G be a finite group and k a field of characteristic p】0.In this paper we consider the support variety for the cohomology module Ext<sub>kG</sub><sup>*</sup>（M,N） where M and N are kG-modules.It is the subvariety of the maximal ideal spectrum of H<sup>*</sup>（G,k） of the annihilator of the cohomology module. For modules in the principal block we show that that the variety is contained in the intersections of the varieties of M and N and the difference between the that intersection and the support variety of the cohomology module is contained in the group theoretic nucleus.For other blocks a new nucleus is defined and a similar theorem is proven.However in the case of modules in a nonprincipal block several new difficulties are highlighted by some examples.
基金Supported by NsF of China (10671160, 10871057), NSF (A200802) PDSF of Heilongjiang Province, China I Supported by NSF of China (10825101)"One Hundred Talents Program" from USTC
文摘In this paper we mainly study the derivations for even part of the finite-dimensional odd Hamiltonian superalgebra HO over a field of prime characteristic. We first give the generating set of the even part g of HO. Then we compute the derivations from g into the even part m of the generalized Witt superalgebra. Finally, we determine the derivation algebra and outer derivation algebra of and the dimension formulas. In particular, the first cohomology groups H^1（g;m） and H^1（g;g） are determined.
基金supported by National Science Foundation of China(Grant Nos.10421101,10721061
文摘In the present note that grew out of my talk given at the conference in honor of Prof. Zhong Tongde,I give a survey of some recent results about holomorphic vector bundles over general Hopf manifolds.
文摘The author introduces a notion of weakIsequences and characterizes such sequences by means of homological methods.This notion extends the notion of weakMsequences and thus extends the notions of generalized Cohen Macaulay modules and Buchsbaum modules to more general cases.
基金Supported by the National Natural Science Foundation of China （grant No. 10371036 11271119） and the Natural Science foundation of Beijing （grant No. 1122002）.Acknowledgement. The authors would like to express their gratitude to the referee（s） for helpful comments and valuable suggestions.
文摘Let A be a connected minimal representation-infinite algebra over an alge- braically closed field k. In this paper, we investigate the simple connectedness and strong simple connectedness of A. We prove that A is simply connected if and only if its first Hochschild cohomology group Hi （A） is trivial. We also give some equivalent conditions of strong simple connectedness of an algebra A.
文摘In this paper we consider the isomorphic classes of H-Galois extensions with the normal basis property over a fixed commutative ring B for a finite Hopf algebra H.The result is that when H is cocommutative,then the above isomorphic classes and the 2-nd cohomology group H^2(L~*,U) are isomorphic as groups under the product defined in this paper,where L=BH,L~*=Hom_B(L,B).
基金supported by the University of Virginia and the Hausdorff Center for Mathematics in Bonn
文摘This paper concerns a certain subcategory of the category of representations for a semisimple algebraic group G in characteristic p, which arises from the semisimple modules for the corresponding quantum group at a p-th root of unity. The subcategory, thus, records the cohomological difference between quantum groups and algebraic groups. We define translation functors in this category and use them to obtain information on the irreducible characters for G when the Lusztig character formula does not hold.
基金supported by National Natural Science Foundation of China(Grant No.11371188)the PAPD(Priority Academic Program Development)of Jiangsu Higher Education Institutions
文摘For a commutative ring R with a unit, an R-homology rose is a topological space whose homology groups with R-coefficients agree with those of a bouquet of circles. In this paper, we study some special properties of the fundamental groups of R-homology roses and their covering spaces, from which we obtain some results supporting the Carlsson conjecture on free(Zp)ractions. In addition, we discuss how to search candidates of the counterexamples of Wall's D(2)-problem among R-homology roses and R-acyclic spaces and propose some candidates.
