弱Hopf量子Yang-Baxter H-模基本定理(英文)

本文引入弱Hopf量子Yang-Baxter模概念.利用弱Hopf模基本定理的方法,获得了弱Hopf量子Yang-Baxter模基本定理,进一步还得到了相关Hopf模基本定理.     

弱Hopf群T-余代数上的弱Doi-Hopf群模

在弱Hopf群T-余代数情形下,弱量子Yetter-Drinfeld群模的概念被引入,并证明了弱量子Yetter-Drinfeld群模是特殊的弱Doi-Hopf群模.接着建立了弱量子Yetter Drinfeld群模范畴与弱Hopf群双余模代数的余不动点子代数B上模范畴之间的伴随对.最后考虑了弱量子Yetter-Drinfeld群模的积分.     

量子群主Tilting模的张量积及其滤过

A=z[υ]Ω,Ω是Z[υ]的由υ-1和奇素数p生成的理想.U是A上的量子代数.令φp是p次分圆多项式,B=A/(φp),Γ是商代数B关于理想(ξ-1)的完备化,式中ξ是p次本原根.对λ∈X+,Mr(λ)表首权为λ的不可分解Uг-Tilting模(称为主Ur模).本文给出了量子群主Ur模的张量积定理.对p≥2(h-1),在p2室中描述了量子群主Ur模好滤过滤过商之首权的分布状态及其滤过重数.作为例子,对秩1型和A2型的量子群情形给出了p2室中一般位置室主Ur模好滤过的分解模式.     

量子群主Tilting模的张量积及其滤过

Ａ＝ Ｚ［ｖ］Ω,Ω Ｚ［ｖ］的由－１和奇ｐ生成的理想． Ｕ是 Ａ上的量子代数．令 фｐ是 ｐ次分圆多项式, Ｂ＝ Ａ／（фｐ）,г 是商代数 Ｂ关于理想（ζ— １）的完备化,式中ζ是ｐ次本原根．对人 λ∈ Ｘ＋, Ｍг（λ）表首权为λ的不可分解 Ｕг－Ｔｉｌｔｉｎｓ模（称为主 Ｕг模）．本文给出了量子群主 Ｕг模的张量积定理．对 ｐ≥ ２（ｈ－１）,在 ｐ２室中描述了量子群主 Ｕг模好滤过滤过商之首权的分布状态及其滤过重数作为例子,对秩１型和 Ａ２型的量子群情形给出了 Ｐ２室中一般位置室主 Ｕг模好滤过的分解模式．     

关于Skew-Hopf对的二重交叉积的表示模

研究了Skew-Hopf对的二重交叉积D（X，A）的表示模，得到了M成为D（X，A）的表示模的充要条件；D（X，A）－模与量子Yang-Baxter X-模、A－模之间的联系也被揭示出来。     

Hopf代数上的扭曲模和(斜)余配对Hopf模

本文第一部分主要把扭曲的方法运用到模上,从而得到扭曲模．作为特例,我们构造了H  M的Smaush模和量子模．当K是有限维Hopf代数,证明K*  M是一个右D(K)-Hopf模,因此得到了一个基本同构定理．第二部分主要把斜余配对双代数进行推广,得到了斜余配对Hopf模,并且给出判断斜余配对Hopf模的一个充要条件．     

U_(r,t)在量子平面上的模代数结构

当q不是单位根,且所在的域是复数域时,给出了U_(r,t)在量子平面上模代数结构的完全分类,并描述了这些表示.有趣的是,在某些情况下,有C_q[x,y]=∞⊕n=0C_q[x,y]_n,其中C_q[x,y]_n是量子平面的n次齐次部分,同构于U_(r,t)的某种不可约模V_(1,n,γ_1)n/2.     

对偶双代数

在文献[1]中,作者引入了扭曲积（twistingproduct）概念,并指出量子偶（Drinfel’ddouble）D（H）为张量积代数的扭曲积．本文把扭曲积加以推广为扭曲模,并给出它的基本结构定理．同时,我们引进对偶Hopf模,它是[2]中对偶Hopf模的发展．     

Smash积代数和量子模范畴中的Hopf代数的新对偶

本文引入模代数的一种新对偶,它推广了代数的有限对偶概念.并证明通过这种新对偶,模代数的对偶为余模余代数,从而形成Smash余积,而且证明了Smash积的对偶是Smash余积,即有(A#H)0≌HA0×H0余代数同构.最后证明量子模范畴中的Hopf代数通过这种新对偶是自对偶的.     

Smash积代数和量子模范畴中的Hopf代数的新对偶

本文引入模代数的一种新对偶,它推广了代数的有限对偶概念,并证明:通过这种新对偶,模代数的对偶为余模余代数,从而形成Smash余积,而且证明了Smash积的对偶是Smash余积,即有(A#H)~0 _HA~0×H~0余代数同构,最后证明量子模范畴中的Hopf代数通过这种新对偶是自对偶的。     

Bipolar theorem for quantum cones

In this note duality properties of quantum cones are investigated. We propose a bipolar theorem for quantum cones, which provides a new proof of the operator bipolar theorem proved by Effros and Webster. In particular, a representation theorem for a quantum cone is proved.     

A representation theorem for quantum systems

In this note representations of quantum systems are investigated. We propose a unital bipolar theorem for unital quantum cones, which plays a key role in proving a representation theorem for quantum systems. It turns out that each quantum system is identified with a certain quantum L^∞-system up to a quantum order isomorphism.     

On the realization of symmetries in quantum mechanics

The aim of this paper is to give a simple, geometric proof of Wigner’s theorem on the realization of symmetries in quantum mechanics that clarifies its relation to projective geometry. Although several proofs exist already, it seems that the relevance of Wigner’s theorem is not fully appreciated in general. It is Wigner’s theorem which allows the use of linear realizations of symmetries and therefore guarantees that, in the end, quantum theory stays a linear theory. In the present paper, we take a strictly geometrical point of view in order to prove this theorem. It becomes apparent that Wigner’s theorem is nothing else but a corollary of the fundamental theorem of projective geometry. In this sense, the proof presented here is simple, transparent and therefore accessible even to elementary treatments in quantum mechanics.     

On the realization of symmetries in quantum mechanics

The aim of this paper is to give a simple, geometric proof of Wigner’s theorem on the realization of symmetries in quantum mechanics that clarifies its relation to projective geometry. Although several proofs exist already, it seems that the relevance of Wigner’s theorem is not fully appreciated in general. It is Wigner’s theorem which allows the use of linear realizations of symmetries and therefore guarantees that, in the end, quantum theory stays a linear theory. In the present paper, we take a strictly geometrical point of view in order to prove this theorem. It becomes apparent that Wigner’s theorem is nothing else but a corollary of the fundamental theorem of projective geometry. In this sense, the proof presented here is simple, transparent and therefore accessible even to elementary treatments in quantum mechanics.     

Quantum systems and representation theorem

In this paper we investigate quantum systems which are locally convex versions of abstract operator systems. Our approach is based on the duality theory for unital quantum cones. We prove the unital bipolar theorem and provide a representation theorem for a quantum system being represented as a quantum $L^{\infty }$ -system.     

On the similarity problem for locally compact quantum groups

A well-known theorem of Day and Dixmier states that any uniformly bounded representation of an amenable locally compact group G on a Hilbert space is similar to a unitary representation. Within the category of locally compact quantum groups, the conjectural analogue of the Day–Dixmier theorem is that every completely bounded Hilbert space representation of the convolution algebra of an amenable locally compact quantum group should be similar to a ?-representation. We prove that this conjecture is false for a large class of non-Kac type compact quantum groups, including all q-deformations of compact simply connected semisimple Lie groups. On the other hand, within the Kac framework, we prove that the Day–Dixmier theorem does indeed hold for several new classes of examples, including amenable discrete quantum groups of Kac-type.     

An operator-valued Lyapunov theorem

We generalize Lyapunov's convexity theorem for classical (scalar-valued) measures to quantum (operator-valued) measures. In particular, we show that the range of a nonatomic quantum probability measure is a weak^?-closed convex set of quantum effects (positive operators bounded above by the identity operator) under a sufficient condition on the non-injectivity of integration. To prove the operator-valued version of Lyapunov's theorem, we must first define the notions of essentially bounded, essential support, and essential range for quantum random variables (Borel measurable functions from a set to the bounded linear operators acting on a Hilbert space).     

GENERALIZED OPERATORS AND P（φ）2 QUANTUM FIELDS

In this paper by Sobolev imbedding theorem and characterization theorem of generalized operators the existence of P(φ)2 quantum fields as generalized operators is obtained and a rigorous mathematical interpretation of renormalization procedure is given under white noise theory.     

伪自伴量子系统的酉演化与绝热定理

经典量子系统的哈密尔顿是自伴算子.哈密尔顿算符的自伴性不仅确保了系统遵循酉演化,而且也保证了它自身具有实的能量本征值.但是,确实有一些物理系统,其哈密尔顿是非自伴的,但也具有实的能量本征值,这种具有非自伴哈密尔顿的系统就是非自伴量子系统.具有伪自伴哈密尔顿的系统是一类特殊的非自伴量子系统,其哈密尔顿相似于一个自伴算子.本文研究伪自伴量子系统的酉演化与绝热定理.首先,给出了伪自伴算子定义及其等价刻画;其次,对于伪自伴哈密尔顿系统,通过构造新内积,证明了伪自伴哈密尔顿在新内积下是自伴的,并给出了系统在新内积下为酉演化的充分必要条件.最后,建立了伪自伴量子系统的绝热演化定理及与绝热逼近定理.