摘要:

Let (A,△) be a Hopf-von Neumann algebra andRbe the unitary fundamental operator onAdefined by Takesaki in [28]:R(a⊗b) = △(b)(a⊗ 1). ThenR12R23=R23R13R12(see lemma 4.9 of [28]). This operatorRplays a vital role in the theory of duality for von Neumann algebras (see [28] or [2]). IfVis a vector space over an arbitrary fieldk, we shall study what we have called the Hopf equation:R12R23=R23R13R12in Endk(V⊗V⊗V). TakingW=rRrthe Hopf equation is equivalent with the pentagonal equation:Wl2Wl3W23=W23W12from the theory of operator algebras (see [2]), whereWare viewed as map inL(K⊗K), for a Hilbert spaceK. For a bialgebraH, we shall prove that the classic category of Hopf modulesplays a decisive role in describing all solutions of the Hopf equation. More precisely, ifHis a bialgebra overkandis anH-Hopf module, then the natural mapR=R(M, ,ρ)a solution for the Hopf equation. Conversely, the main result of this paper is a FRT type theorem:ifMis a finite dimensional vector space andR∈ Endk-(M⊙M) is a solution for the Hopf equation, then there exists a bialgebraB(R) such thatBy applying this result, we construct now examples of noncommutative and noncocomimitative bialgebras which are different from the ones arising from quantum group theory. In particular, over a field of characteristic two, an example of five dimensional noncommutative and noiicocommutative bialgebra is given.

ISSN:0092-7872    

DOI:10.1080/00927879808826329

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1998

数据来源: Taylor

摘要:

LetMbe ak-vector space andR∈ Endk(M⊗M)In [13] we introduced and studied what we called the Hopf equation:R12R23=R23R13R12.By means of a FRT type theorem, we have proven that the categoryofH-Hopf modules is deeply involved in solving this equation. In the present paper, we continue to study the Hopf equation from another perspective:having in mind the quantum Yang-Baxter equation, in the solution of which the co-quasitriangular (or braided) bialgebras play an important role (see [8]), we introduce and study what we call bialgebras with Hopf functions. The main theorem of this paper shows that, ifMis finite dimensional, any solutionRof the Hopf equation has the form R = Rσ. whereMis a right comodule over a bialgebra with a Hopf function (B{R),C,σ) andRσis the special map.

ISSN:0092-7872    

DOI:10.1080/00927879808826330

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1998

数据来源: Taylor

ISSN:0092-7872    

DOI:10.1080/00927879808826331

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1998

数据来源: Taylor

ISSN:0092-7872    

DOI:10.1080/00927879808826332

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1998

数据来源: Taylor

摘要:

Given a maximal subgroupMof a finite groupG,a θ completion ofMinGis any subgroupCsuch thatCis not contained inMwhileMG, the core ofMinG, is contained inCandC/MGhas no propor normal subgroup ofG/MG. By using this concept we can reveal the relationship between the concepts of completions and θ-pairs introduced respectively by Deskins, Mukherjee and Bhattacharya. The concept of maximal θ-completions offers a convenience for us to study the Deskins completions in inductions. We obtain in this paper several results which imply a group to be solvable, supersolvable and nilpotent.

ISSN:0092-7872    

DOI:10.1080/00927879808826333

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1998

数据来源: Taylor

摘要:

This paper is a continuation of the paper “On the Deskins completions, theta completions and theta pairs for maximal subgroups ”.In the former paper, Zhao Yaoqing introduced the concept of θ-completions associated to a maximal subgroup of a finite group. The concept offers a convenience for us to study the completions introduced by Deskins and gives us a way to reveal the relationship between the concepts of completions and θ-pairs, the latter concept is introduced by Mukherjee and Bhattacharya. The present paper is devoted to discussing the π-solvability, π-supersolvability and π-nilpotency of a finite group by using the θ-completions. Moreover, a new proof on the Deskins conjecture concerning the supersolvability is included.

ISSN:0092-7872    

DOI:10.1080/00927879808826334

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1998

数据来源: Taylor

摘要:

We compute the Euler obstruction and Mather’s Chern class of the discriminant hypersurface of a very ample linear system on a nonsingular variety. Comparing the codimension-1 and 2 terms of this and other characteristic classes of the discriminant leads to a quick computation of the degrees of the loci of cuspidal and binodal sections of a very ample line bundle on a smooth variety, and of the, tacnodal locus for linear systems on a surface. We also compute explicitly all terms in the Schwartz-MacPherson’s classes of strata of the discriminant in the P9of of cubic plane curves, and of the discriminant of∣O(d)∣ on P1.

ISSN:0092-7872    

DOI:10.1080/00927879808826335

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1998

数据来源: Taylor

ISSN:0092-7872    

DOI:10.1080/00927879808826336

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1998

数据来源: Taylor

ISSN:0092-7872    

DOI:10.1080/00927879808826337

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1998

数据来源: Taylor

摘要:

We define an integral domainDto be anti-Archimedean if. For example, a valuation domain or SFT Prüfer domain is anti-Archimedean if and only if it has no height-one prime ideals. A number of constructions and stability results for anti-Archimedean domains are given. We show thatDis anti-Archimedeanis quasilocal and in this caseis actually ann-dimensional regular local ring. We also show that ifDis an SFT Prüfer domain, thenis a Krull domain for any set of indeterminates {Xα}.

ISSN:0092-7872    

DOI:10.1080/00927879808826338

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1998

数据来源: Taylor