THE JUMPING PHENOMENON OF THE DIMENSIONS OF COHOMOLOGY GROUPS OF TANGENT SHEAF

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.

Cohomology groups of a new class of Kadison-Singer algebras

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.

COMPLETELY BOUNDED COHOMOLOGY OF NON-SELFADJOINT OPERATOR ALGEBRAS

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.

FINITE p-GROUPS WHICH CONTAIN A SELF-CENTRALIZING CYCLIC NORMAL SUBGROUP

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.

Cohomology of the Schrdinger Algebra S(1)

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.

Low Dimensional Cohomology of Hom-Lie Algebras and q-deformed W(2, 2) Algebra

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.

Derivations and Automorphisms of the Positive Part of the Two-parameter Quantum Group U_(r,s)(B_3)

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）.

Derivations of the Positive Part of the Two-parameter Quantum Group of Type G_(2)

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.

Varieties for Cohomology with Twisted Coefficients

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 ExtkG*（M,N） where M and N are kG-modules.It is the subvariety of the maximal ideal spectrum of H*（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.

Derivations of the Even Part of the Odd Hamiltonian Superalgebra in Modular Case

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.

TOWARD A THEORY OF WEAK I SEQUENCES

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.

On Simple Connectedness of Minimal Representation-Infinite Algebras

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.

On the Isomorphic Classes of Hopf-Galois Extensions with Normal Bases over a Commutative Ring

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).