Cohomological properties of the quantum shuffle product and application to the construction of quasi-Hopf algebras

For a commutative algebra the shuffle product is a morphism of complexes. We generalize this result to the quantum shuffle product, associated to a class of non-commutative algebras (for example all the Hopf algebras). As a first application we show that the Hochschild-Serre identity is the dual statement of our result. In particular, we extend this identity to Hopf algebras. Secondly, we clarify the construction of a class of quasi-Hopf algebras.     

Multiderivations of Coxeter arrangements

Let V be an ℓ-dimensional Euclidean space. Let G⊂O(V) be a finite irreducible orthogonal reflection group. Let ? be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H∈? choose α _H ∈V ^* such that H=ker(α _H ). For each nonnegative integer m, define the derivation module D ^(m) (?)={θ∈Der _S |θ(α _H )∈Sα ^m _H }. The module is known to be a free S-module of rank ℓ by K. Saito (1975) for m=1 and L. Solomon-H. Terao (1998) for m=2. The main result of this paper is that this is the case for all m. Moreover we explicitly construct a basis for D ^(m) (?). Their degrees are all equal to mh/2 (when m is even) or are equal to ((m−1)h/2)+m _i (1≤i≤ℓ) (when m is odd). Here m ₁≤···≤m _ℓ are the exponents of G and h=m _ℓ+1 is the Coxeter number. The construction heavily uses the primitive derivation D which plays a central role in the theory of flat generators by K. Saito (or equivalently the Frobenius manifold structure for the orbit space of G). Some new results concerning the primitive derivation D are obtained in the course of proof of the main result. Oblatum 27-XI-2001 & 4-XII-2001?Published online: 18 February 2002     

The homotopy categorical crossed module of a CW-complex

Crossed modules have longstanding uses in homotopy theory and the cohomology of groups. The corresponding notion in the setting of categorical groups, that is, categorical crosses modules, allowed the development of a low-dimensional categorical group cohomology. Now, its relevance is also shown here to homotopy types by associating, to any pointed CW-complex (X,∗), a categorical crossed module that algebraically represents the homotopy 3-type of X.     

Thermoelectric properties of tin clathrates under hydrostatic pressure

We report the temperature dependence of electrical resistance (R) and thermopower (S) of clathrate Cs₈Sn₄₄ under high pressure up to 1.2 GPa. We observe a reversible gap widening, prominent relaxation effect of R, irreversible increase of |S| under high pressure. We also find that the power factor S²σ (σ: electrical conductivity) reaches a maximum at pressure of 0.3 GPa. Comparison of the experimental results with band structure calculations suggests that the intrinsic vacancy in the clathrate structure of Cs₈Sn₄₄ plays an important role in transport properties under high pressure. Measurements on Cs₈Zn₄Sn₄₂, a clathrate which has defects other than vacancies, are compared with Cs₈Sn₄₄. The results indicate that replacing Sn by Zn has similar effect as the intrinsic vacancy on S.     

On the Log-Laplace Equation for Nonlocal Operators Generating Sub-Markovian Semigroups

We consider the semilinear Cauchy problem for a class of pseudo-differential operators generating sub-Markovian semigroups. Solutions of such problems with negative definite nonlinearity play an important role in constructing branching measure-valued processes. We establish local existence and uniqueness of solutions in the context of the Dirichlet space associated to the problem. Comparison and global properties of solutions are also studied. Accepted 29 August 2001. Online publication 17 December 2001.     

Finite Groups Acting on Homology 3-Spheres: On a Result of M. Reni

 We give a list including all finite groups G which admit smooth orientation preserving actions on homology 3-spheres (arbitrary actions, i.e. possibly with fixed points; if the action is free then the group G has periodic cohomology and the classification of such groups is well known). The main work in this direction is due to M. Reni. In the present paper, we complete and extend his results for the case of nonsolvable groups G. Received 19 March 2001; in revised form 15 September 2001     

Researches of DC—Problem Under the Model S in which the Numbers of Defective Coins are not Fixed

In this paper,we research general Defective Cion Problem under the model S,i.e,the number d of Defective cions is not fixed. For d=O,1,or 2. we get some good results.     

The quantum de Rham complexes associated with SL h (2)

Quantum de Rham complexes on the quantum plane and the quantum group itself are constructed for the nonstandard deformation of Fun(SL(2)). It is shown that in contrast to the standardq-deformation of SL(2), the above complexes are unique for SL _h (2). Also, as a byproduct, a new deformation of the two-dimensional Heisenberg algebra is obtained which can be used to construct models ofh-deformed quantum mechanics.     

系列类立方烷型Fe4S4簇合物的XPS分析

对7种类立方烷型Fe_4S_4簇合物进行了XPS分析,得到了中心离子Fe的价态,给出了系列Fe_4S_4中骨架硫的结合能,由峰合成结果给出各种S、N的比例与按结构式计算的结果相同。