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物理专业英语词汇 Q 物理专业英语词汇q物理专业英语词汇(q)qbranch q分支qmeter q值计qnumber q数qswitch q开关qswitchedlaser q控制器激光器qvalue q值quadrantelectrometer 象限静电计quadraticform 二次形式quadrupole 四极quadrupolecoupling 四极耦合quadrupoledeformation 四极畸变quadrupoleinteraction 四极相互酌quadrupolemagnet 四极磁铁quadrupolemoment 四极矩quadrupolemomentum 四极矩quadrupolequadrupoleforce 四极四极力quadrupoleradiation 四极电磁辐射quadrupoleresonance 四极共振quadrupoletransition 四极光子quadrupolevibration 四极振荡qualitativeanalysis 定性分析quantacon 光电倍增管quantitativeanalysis 定量分析quantitativespectralanalysis 定量光谱分析 quantity 量quantityofelectricity 电量quantityofflow 量thermocouplepyrometer热电偶高温计quantityofheat 热量quantityoflight 光量quantityofmotion 动量quantityofstate 状态量quantization 量子化quantizedfield 量子场quantizedsystem 量子化系统quantizedvortex 量子旋涡quantumanomaly 量子反常quantumbiology 量子生物学quantumchemistry 量子化学quantumchromodynamics 量子色动力学quantumcondition 量子条件quantumcosmology 量子宇宙学quantumcreationoftheuniverse 宇宙的量子产生 quantumdefect 量子筐quantumdefecttheory 量子筐理论quantumdisorderedsystem 量子无序系quantumefficiency 量子产额quantumelectrodynamics 量子电动力学quantumelectronics 量子电子学quantumenergy 量子能量quantumfield 量子场quantumfieldtheory 量子场论quantumfluctuation 量子涨落quantumfluid 量子铃quantumgravitation 量子引力quantumgravitationalfluctuation 量子引力差值 quantumgravity 量子引力quantumgroup 量子群quantumhalleffect 量子霍尔效应quantumhilbertspace 量子希耳伯空间quantumhypothesis 量子假说quantuminversescatteringmethod 量子逆反射法 quantumjump 量子跃迁quantumlatticemodel 量子图形模型quantumliquid 量子液体quantumlogic 量子逻辑quantummechanics 量子力学quantummontecarlomethod 量子蒙特卡罗法quantumnoise 量子噪声quantumnumber 量子数quantumoptics 量子光学quantumorbit 量子轨道quantumphysics 量子物理学quantumsizeeffect 量子尺寸效应quantumsolid 量子液态quantumsoliton 量子孤立子quantumstate 量子状态quantumstatisticalmechanics 量子统计力学 quantumstatistics 量子统计数据quantumtheory 量子论quantumtheoryoffield 量子场论quantumtheoryofgravity 引力场的量子论quantumtransition 量子光子quantumwell 量子势阱quantumwellopticalwaveguide 量子势阱光波导 quantumyield 量子产额quantumbrowwidth18fieldeng 量子quark 夸克quarkatom 夸克偶素quarkcondensate 夸克凝聚quarkconfinement 夸克禁锢quarkflavor 夸克味quarkgluonplasma 夸克胶子等离子体quarkleptonsymmetry 夸克轻子对称quarkmodel 夸克模型quarkstar 夸克星quarkonium 夸克偶素quarterwaveplate 四分之一波片quartet 四重态quartz 水晶quartzclock 石英钟quartzfibreelectroscope 石英丝验电器quartzglass 石英玻璃quartzmonochromator 水晶单色仪quartzoscillator 石英振荡器quartzplate 水晶片quasar 类星体thermocouplepyrometer热电偶高温计quasicrystal 科东俄晶体quasielasticforce 准弹性力quasielasticscattering 科东俄弹性散射quasiergodichypothesis 准脯历经假说quasifouriertransformhologram 科东俄傅里叶转换全息图 quasimolecularresonance 准分子共振quasimonochromaticlight 科东俄单色光quasistaticporcess 准静态过程quasistationaryelectriccurrent 准稳电流quasistationarystate 准稳态quasistellarobject 类星体quasistellarsource 类星射电源quasiviscouseffect 科东俄表面张力效应 quasianisotropy 类蛤异性quasimode 准模quasimolecule 准分子quasiparticle 科东俄粒子quasiparticletunnelling 准粒子隧道效应 quenching 点燃quenchingcircuit 熄灭电路quiescentprominence 宁静日珥quietsun 宁静太阳quintet 五重态。
quadrupole probe,四极探头quadrupole residual gas analyzer,四极残余气体分析器quadrupole rods,四极杆qualification,鉴定qualification standard,鉴定标准qualification test,鉴定试验qualitative analysis,定性分析quantitative differential thermal analysis (QDTA),定量差示热分析qualitative physical model,定性物理模型quality,质量quality assurance,质量保证quality control (QC),质量控制quality inspection,质量检验quality management,质量管理quality supervision,质量监督quantitative analysis,定量分析quantitative differential thermal analysis,定量差热分析quantitative differential thermal analyzer,定量差热（分析）仪quantity of heat,热量quantity of illumination,光照量quantity of radiant energy,辐射能量quantity of radiation,幅射能量quantity to be measured,被测量quantization,量化quantization error,量化误差quantized error,量化误差quantized noise,量化噪声quantized signal,量化信号quartet,四重布置；四重检测quarter circle orifice plate,1/4圆孔板quartz-Bourdon tube (pressure)gauge,石英弹簧管压力计quartz crystal unit humidity transducer[sensor]晶体振子式湿度传感器quartz module,石英组件quartz spring gravimeter,石英弹簧重力仪quartz thermometer,石英温度计；石英温度表quasi-equilibrium theory,准平衡理论quasi-molecular ion,准分子离子quasi-periodic vibration,准周期振动quasi-rigid rotor,准刚性转子quasi-sinusoidal quantity,准正弦量quasi-static unbalance,准静不平衡quasilinear chareacteristics,准线性特性quenching,阻塞question and answer mode,问答式queuing theory,排队论quick thermophysical property measuring apparatus,热物性快速测定仪quintet,五重布置；五重检测quotient-meter,商值表Rrack,框架radar altimeter,雷达高度计radial clearance,径向间隙radial distortion,径向畸变radial electrostatic field analyzer,径向静电场分析器radial heat shield,隔热屏radiance,幅射亮度；辐射率radiance temperature,亮度温度radiance thermometry,亮度测温法radiant element,辐射元件radiant energy,辐射能（量）radiant flux,辐射通量radiant intensity,辐射强度radiant power,辐射功率radiation,辐射radiation balance meter,辐射平衡表radiation detecting device,辐射检测器radiation dose transducer [sensor],射线齐量传感器radiation energy,辐射能量radiation exitance,辐（射）出（射）度radiation fin bonnet,散热片型上阀盖radiation flux,辐射通量radiation heatflowmeter,辐射热流计radiation intensity,辐射强度radiation monitor,辐射监测仪radiation pyrometer,辐射高温计radiation sensor,射线敏感元件radiation temperature,辐射温度radiation temperature transducer [sensor],辐射温度传感器radiation test,辐射试验radiation thermometer,辐射温度计radiation thermometry,辐射测温法radiation transducer [sensor],射线传感器radio chromatography,放射色谱法radio frequency sensor,射频敏感器radio wave borehole penetration system,井中无线电波仪radio wave penetration system,无线电波透视仪radioactive materials packaging,放射性物质包装radioactive survey,放射性测量radiodirection finder,无线电方位测定器radiometer,辐射仪；辐射表radiosonde,无线电探空仪radiotheodolite,无线电经纬仪radiowave penetration instrument,无线电波透视法仪器rain proof instrument,防溅式仪器仪表rainfall amount,雨量rainfall intensity meter,雨量强度计rainfallmeter,雨量强度计raingauge,雨量器raingauge receiver,承水器raingauge shield,雨量器防风罩Raman line,拉曼谱线Raman shift,拉曼位移Raman spectrum,拉曼光谱ramp function,斜坡函数ramp response,斜坡响应ramp response time,斜坡响应时间random access,随机存取random disturbance,随机扰动random error,随机误差random event,随机事件random failure,偶然失效random noise,随机噪声；无规噪声random process,随机过程random signal,随机信号random uncertainty,随机不确定度random vibration,随机振动range,范围range-change device,范围转换器；量限转换器range of load,负荷范围range of strain,应变范围range of stress,应力范围range response time,范围响应时间rangefinder,测距仪ranging distance,测程Rankine temperature scale,兰金温标rangeability,范围度rapid Fourier transform NMR correlation spectroscopy,快速傅立叶变换核磁共振相关波谱法rapid TG-DTA unit,快速加热差示热分析仪/热天平rate of change limiting control,变化率极限控制rate of deformation,变形速率rate of loading,加荷速率rate of salt spray precipitation,盐雾沉降率rate time,预调时间rated accuracy limit primary current (of a protective current transformer),（保护用电流互感器的）准确度限定的额定一次电流。
the intensional qualification ofquantificationIntensional qualification of quantification refers to the idea that the meaning of a quantified statement depends not only on the objects being quantified but also on the properties or qualities associated with those objects. In other words, the meaning of a statement like "all dogs have fur" is not just a matter of counting all the dogs and observing their fur, but also involves an understanding of what it means to be a dog and what it means to have fur.One example of intensional qualification in action can be seen in the statement "all bachelors are unmarried." The meaning of this statement depends not just on the fact that there are a certain number of unmarried men in the world, but also on our understanding of what it means to be a bachelor. If we were to define bachelor as "a man who has never been married and has no children," then the statement "all bachelors are unmarried" would be tautological and unsurprising. But if we were to define bachelor in adifferent way, such as "a man who is over 30 and has never been married," then the statement would be false, since there may well be unmarried men over 30 who are not considered bachelors.Intensional qualification can also come into play in more complex statements. For example, consider the statement "some cats are not black." The truth of this statement depends not just on the existence of non-black cats, but alsoon our understanding of what it means to be a cat and what it means to be black. If we were to define cat as "a small,furry animal that meows," and black as "a color associated with darkness," then the statement would be true, since there are certainly cats that are not black. But if we define cat more narrowly as "a member of the Felidae family," and black more broadly as "any color that is not white," then the statement might be false, as all members of the Felidae family are typically black, brown, or orange.Overall, intensional qualification of quantification highlights the importance of context and interpretation in understanding the true meaning of statements involving quantifiers. By considering not just the objects being quantified, but also the qualities or properties associated with those objects, we can arrive at a more nuanced and accurate understanding of the world around us.。
反量化英语Quantification has become an increasingly prevalent aspect of modern life, with data and metrics being used to measure and evaluate various facets of our existence. While this trend has brought about numerous benefits, it has also given rise to a growing concern regarding the potential drawbacks of an over-reliance on quantification. In this essay, we will explore the concept of "anti-quantification" and examine the arguments for a more balanced approach to the role of data and metrics in our lives.One of the primary arguments against the excessive use of quantification is the inherent reductionism inherent in the process. By reducing complex phenomena to numerical values, we risk oversimplifying the nuances and contextual factors that contribute to the richness and depth of human experience. This can lead to a distorted understanding of reality, where the quantifiable aspects are prioritized at the expense of the qualitative and intangible elements that are equally, if not more, important.Moreover, the reliance on quantification can foster a culture of obsession with metrics and a fixation on numerical targets, often at the expense of deeper, more meaningful goals. This can manifest invarious domains, from education, where test scores become the primary measure of success, to healthcare, where patient outcomes are reduced to a series of statistics. In such scenarios, the true purpose of these institutions – to nurture well-rounded individuals and promote holistic well-being – can become obscured.Another concern with the overuse of quantification is the potential for unintended consequences and the distortion of behavior. When individuals and organizations are evaluated primarily based on numerical targets, they may be tempted to manipulate or game the system in order to achieve those targets, even if it means sacrificing integrity or ethical considerations. This can lead to a culture of mistrust, where the reliability and validity of the data become increasingly questionable.Furthermore, the reliance on quantification can contribute to a sense of dehumanization, where individuals are reduced to mere data points, stripped of their unique experiences, emotions, and personal narratives. This can have profound implications for our social interactions, decision-making processes, and the way we perceive and value one another.In response to these concerns, the concept of "anti-quantification" advocates for a more balanced and nuanced approach to the use of data and metrics. This perspective recognizes the value ofquantification in certain contexts, such as scientific research, policy-making, and decision-support systems, but also acknowledges the need to temper its application with a deeper understanding of the qualitative and contextual factors that shape human experiences and societal dynamics.At the heart of the anti-quantification movement is a call for a greater emphasis on the subjective, the experiential, and the intangible aspects of life. This includes a focus on narrative, storytelling, and the exploration of the human condition through the arts, humanities, and social sciences. By embracing these alternative modes of understanding, we can cultivate a richer, more nuanced perspective on the world around us, one that acknowledges the inherent complexity and diversity of human experiences.Moreover, the anti-quantification approach encourages a more critical and reflective stance towards the use of data and metrics. This involves questioning the underlying assumptions, methodologies, and potential biases that shape the collection and interpretation of data, as well as a willingness to challenge the dominant narratives and preconceptions that often drive the quantification agenda.In practical terms, the implementation of anti-quantification principles could involve a range of strategies, such as the incorporation of qualitative assessments alongside quantitativemeasures, the emphasis on contextual factors in decision-making processes, and the fostering of interdisciplinary collaborations that bridge the divide between the quantitative and the qualitative.Ultimately, the call for anti-quantification is not a rejection of the value of data and metrics, but rather a recognition of the need to strike a balance between the quantifiable and the intangible, the objective and the subjective, in order to create a more holistic and meaningful understanding of the human experience. By embracing this approach, we can work towards a future where the richness and complexity of our lives are not reduced to mere numbers, but instead celebrated and understood in all their nuanced glory.。
a rX iv:mat h /33183v2[mat h.QA ]26Ma y24FLABBY STRICT DEFORMATION QUANTIZATIONS AND K -GROUPS HANFENG LI Abstract.We construct examples of flabby strict deformation quantizations not preserving K -groups.This answers a question of Rieffel negatively.1.Introduction In the passage from classical mechanics to quantum mechanics,one replaces smooth functions on symplectic manifolds (more generally,Poisson manifolds)by operators on Hilbert spaces,and replaces the Poisson bracket of smooth functions by commutators of operators.Thinking of classical mechanics as limits of quantum mechanics,one requires that the Poisson brackets becomes limits of commutators.There is an algebraic way of studying such process using formal power series,called deformation quantization [1,13].In order to study it in a stricter way,Rieffel introduced [6]strict deformation quantization of Poisson manifolds,within the framework of C ∗-algebras.He showed that noncommutative tori arise naturally as strict deformation quantizations of the ordinary torus in the direction of certain Poisson bracket.After that,a lot of interesting examples of strict deformation quantizations have been constructed.See [8,9]and the references therein.We refer the reader to [2,Sections 10.1–10.3]for the basic information about continuous fields of C ∗-algebras.Recall the definition of strict deformation quanti-zation [6,9]:Definition 1.1.[9,Definition 1]Let M be a Poisson manifold,and let C ∞(M )be the algebra of C -valued continuous functions on M vanishing at ∞.By a strict deformation quantization of M we mean a dense ∗-subalgebra A of C ∞(M )closed under the Poisson bracket,together with a continuous field of C ∗-algebras A over a closed subset I of the real line containing 0as a non-isolated point,and linear maps π :A →A for each ∈I ,such that(1)A 0=C ∞(M )and π0is the canonical inclusion of A into C ∞(M ),(2)the section (π (f ))is continuous for every f ∈A ,(3)for all f,g ∈A we havelim →0 [π (f ),π (g )]/(i )−π ({f,g }) =0,(4)π is injective and π (A )is a dense ∗-subalgebra of A for every ∈I .If A ⊇C ∞c (M ),the space of compactly supported C -valued smooth functions on M ,we say that the strict deformation quantization is flabby .2HANFENG LICondition(4)above enables us to define a new∗-algebra structure and a new C∗-norm on A at each by pulling back the∗-algebra structure and norm of π (A)⊆A to A viaπ .Condition(2)means that this deformation of the∗-algebra structure and norm on A is continuous.Given a strict deformation quantization,a natural question is whether the de-formed C∗-algebras A have the same”algebraic topology”,in particular,whether they have isomorphic K-groups.Rieffel’s quantization of Poisson manifolds in-duced from actions of R d[7]and many other examples[5]are known to preserve K-groups.Rieffel showed examples of non-flabby strict deformation quantizations not preserving K-groups,and asked[9,Question18]:Are the K-groups of the de-formed C∗-algebras of anyflabby strict deformation quantization all isomorphic?A nice survey of various positive results on related problems may be found in[10].Shim[11]showed that above question has a negative answer if one allows orb-ifolds.But it is not clear whether one can adapt the method there to get smooth examples.Rieffel also pointed out[9,page321]that in any strict deformation quantization of a non-zero Poisson bracket if one reparametrizes by replacing by 2one obtains a strict deformation quantization of the0Poisson bracket.Thus to answer Rieffel’s question it suffices to consider strict deformation quantizations of the0Poisson bracket.The main purpose of this paper is to answer above question.In Section2we give a general method of constructingflabby strict deformation quantization for the0 Poisson bracket.In particular,we proveTheorem1.2.Let M be a smooth manifold with dim M≥2,equipped with the0 Poisson bracket.If dim M is even(odd,resp.),then for any integers n0≥n1≥0 (n1≥n0≥0resp.)there is aflabby strict deformation quantization{A ,π } ∈I of M over I=[0,1]with A=C∞c(M)such that K i(A )∼=K i(C∞(M))⊕Z n i for all0< ≤1and i=0,1.Theorem1.2is far from being the most general result one can obtain using our construction in Section2.However,it illustrates clearly that a lot of manifolds equipped with the0Poisson bracket haveflabby strict deformation quantizations not preserving K-groups.In order to accommodate some other interesting examples such as Berezin-Toeplitz quantization of K¨a hler manifolds,Landsman introduced a weaker notion strict quantization[3,Definition II.1.1.1][9,Definition23].This is defined in a way similar to a strict deformation quantization,but without requiring the condition(4) in Definition1.1.Ifπ is injective for each ∈I we say that the strict quantization is faithful.It is natural to ask for the precise relation between strict quantizations and strict deformation quantizations.Rieffel also raised the question[9,Question 25]:Is there an example of a faithful strict quantization such that it is impossible to restrictπ to a dense∗-subalgebra B⊆A to get a strict deformation quantization of M?Adapting our method in Section2we also give such an example for every manifold M equipped with the0Poisson bracket.In[4]strict quantizations are constructed for every Poisson manifold,and it is impossible to restrict the strict quantizations constructed there to dense∗-subalgebras to get strict deformation quantizations unless the Poisson bracket is0[4,Corollary5.6].Thus we get a complete answer to Rieffel’s question.FLABBY STRICT DEFORMATION QUANTIZATIONS AND K-GROUPS3 Acknowledgments.I am grateful to Marc Rieffel for many helpful discussions and suggestions,and I also thank the referee for pointing out the reference[10].2.Strict deformation quantizations for the0Poisson bracketWe start with a general method of deforming a C∗-algebra.Let A be a C∗-algebra and A⊆A a dense∗-subalgebra.Let I(A)={b∈M(A):bA,Ab⊆A} be the idealizer of A in the multiplier algebra M(A)of A.Then I(A)is a∗-algebra containing A as an ideal,and for every b∈(I(A))sa clearly bAb is a∗-subalgebra of A.If furthermore the multiplication by b is injective on A,that is, b∈Ann:={b′∈M(A):b′a=0for some0=a∈A},then we can pull back the multiplication and norm on bAb to define a new multiplication×b and a new norm · b on A via the bijection A→bAb.Explicitly,a×b a′=ab2a′anda b= bab .The completion of(A,×b, · b)is isomorphic tob x Ab x at x∈X,as a subfield of the trivial continuousfield of C∗-algebras over X withfibres A,and it contains(b x ab x) as a continuous section for every a∈A.Now we specialize to the commutative case.Let M be a smooth manifold,and let A=C∞(M),A=C∞c(M).Then M(A)is the space C b(M)consisting of all C-valued bounded continuous functions on M,and the strict topology on C b(M)is determined by uniform convergence on every compact subset of M.The idealizer I(A)is the space C∞b(M)consisting of all C-valued bounded smooth functions on M.Given b∈I(A),it is not in Ann exactly if the zero set Z b of b is nowhere dense.Clearly C∞(M\Z b)⊇bAb⊇C∞c(M\Z b),and hence4HANFENG LI158]for every0< ≤1and i=0,1.Thus when n is odd K i(A )∼=K i(C∞(R n)) for all0< ≤1and i=0,1(see for instance[12,page123]for the K-groups of R n and S n).When n is even,K1(A )∼=K1(C∞(R n))for all0< ≤1.When M is compact,in Proposition2.1the element b has to be invertible in C(M)for small and consequently A =C(M).Thus in order to construct strict deformation quantizations for compact M such that the K-groups of A are not isomorphic to those of C(M)for any =0,we have to modify the construction in Proposition2.1.Notice that if we setπ′ (a+λ)=b ab +λfor a∈C∞c(R n),λ∈C in Example2.2,then we get a strict deformation quantization of S n equipped with the0Poisson bracket.This leads to Proposition2.4below.Notation2.3.We denote by F m the space of smooth real-valued functions F on R m such that F is equal to1outside a compact subset of R m and the zero set Z F of F is nowhere dense.Proposition2.4.Let M be a smooth manifold equipped with the0Poisson bracket. Let U be an open subset of M with a diffeomorphismϕ:U→R m.For any F∈F m there is aflabby strict deformation quantization{A ,π } ∈I of M over I=[0,1]with A=C∞c(M)such that A ∼=C∞(M/Y)for every0< ≤1,where Y=ϕ−1(Z F∪{0}).Proof.Set F0=1and F (x)=F(x/ )for all0< ≤1and x∈R m.Then F ∈F m for each ∈I and we can extend the pull-back F ◦ϕ∈C∞(U)to a smooth function b on M by setting it to be1outside U.Clearly b A′b is a∗-subalgebra of A′.Notice that there is a compact set W⊂U such that b =1on M\W for all ∈I,and W containsϕ−1(0).Take an H∈(C∞c(M))R such that H=1 on W.Denote by A′the space of functions in C∞c(M)vanishing atϕ−1(0).Then C∞c(M)=A′⊕C H as complex vector spaces,and H2−H=b (H2−H)b ∈b A′b . It is easy to see that b A′b +C H is a∗-subalgebra of C∞c(M)and the linear map π :C∞c(M)→b A′b +C H defined byπ (a′+λH)=b a′b +λH for a′∈A′andλ∈C is bijective.For each a′∈A′clearly the map →b a′b ∈C∞(M)is continuous on I=[0,1].Thus for each a∈A=C∞c(M),(π (a))is a continuous section in the continuous subfield{A =b A′b +C H=b A′b +C H∼=C∞(M/Y)as desired.Next we describe a case in which we can relate the K-groups of C∞(M/Y)to those of C∞(M)easily:Lemma 2.5.Let D be the subset of R m consisting of points(x1,···,x m)with 0<x1,···,x m<1.Let M,ϕ,F and Y be as in Proposition2.4.Suppose that ∂D⊆Z F⊆¯D.ThenK i(C∞(M\Y))∼=K i(C∞(M))⊕K i(C∞(D\Z F))for i=0,1.FLABBY STRICT DEFORMATION QUANTIZATIONS AND K-GROUPS5 Proof.Letφ:M→M/Y be the quotient map,and let W=φ(M\ϕ−1(D)).Then W is a closed subset of M/Y,and the complement is homeomorphic to D\Z F. Define a mapψ:M/Y→W as the identity map on W andψ((M/F)\W)=φ(Y). Thenψis continuous and proper,i.e.the inverse image of every compact subset of W is compact.Thus the exact sequence0→C∞(D\Z F)→C∞(M/Y)→C∞(W)→0splits.Therefore K i(C∞(M\Y))∼=K i(C∞(W))⊕K i(C∞(D\Z F))for i=0,1. Now Lemma2.5follows from the fact that W is homeomorphic to M.Notice that if a compact set Z⊆R m is the zero set of some non-negative f∈C∞(M),then it is also the zero set of some F∈F m(for instance,take a non-negative g∈C∞c(M)with g|Z=1and set F(x)=f(x)6HANFENG LI[5]G.Nagy,Deformation quantization and K-theory.In:Perspectives on Quantization,(SouthHadley,MA,1996),111–134,Contemp.Math.,214,Amer.Math.Soc.,Providence,RI,1998.[6]M.A.Rieffel,Deformation quantization of Heisenberg m.Math.Phys.122(1989),no.4,531–562.[7]M.A.Rieffel,K-groups of C∗-algebras deformed by actions of R d.J.Funct.Anal.116(1993),no.1,199–214.[8]M.A.Rieffel,Quantization and C∗-algebras.In:C∗-algebras:1943-1993(San Antonio,TX,1993),66–97,Contemp.Math.,167,Amer.Math.Soc.,Providence,RI,1994.[9]M. A.Rieffel,Questions on quantization.In:Operator algebras and operator theory(Shanghai,1997),315–326,Contemp.Math.,228,Amer.Math.Soc.,Providence,RI,1998.arXiv:quant-ph/9712009.[10]J.Rosenberg,Behavior of K-theory under quantization.In:Operator algebras and quantumfield theory(Rome,1996),404–415,Internat.Press,Cambridge,MA,1997.[11]J.K.Shim,A negative answer to a Rieffel’s question on the behavior of K-groups understrict deformation quantization.J.Geom.Phys.44(2003),no.4,475–480.[12]N.E.Wegge-Olsen,K-theory and C∗-algebras.A Friendly Approach.Oxford Science Pub-lications.The Clarendon Press,Oxford University Press,New York,1993.[13]A.Weinstein,Deformation Quantization.S´e minaire Bourbaki Vol.1993/94.Ast´e risque No.227(1995),Exp.No.789,5,389–409.Department of Mathematics,University of Toronto,Toronto ON M5S3G3,CANADA E-mail address:hli@。
a r X i v :m a t h /9802102v 3 [m a t h .D G ] 2 F eb 1999Connes’tangent groupoid and strictquantizationJos´e F.Cari˜n ena a ,Jes´u s Clemente-Gallardo a ,Eduardo Follana a ,Jos´e M.Gracia-Bond´ıa a ,b ,Alejandro Rivero a ,and Joseph C.V´a rilly c a Departamento de F´ısica Te´o rica,Universidad de Zaragoza,50009Zaragoza,Spain b Departamento de F´ısica,Universidad de Costa Rica,2060San Pedro,Costa Rica c Departamento de Matem´a ticas,Universidad de Costa Rica,2060San Pedro,Costa Rica 1Introduction1.1MotivationRecently,Rieffel has published a list of open problems [1]in quantization.The main aim of this paper is to address Question 20of Rieffel’s list:“...in what ways can a suitable Riemannian metric on a manifold M be used to obtain a strict deformation quantization of T ∗M ?”We do that by giving proofs for (a slightly improved variant of)a construction sketched by Connes in Section II.5of his book [2]on noncommutative geometry,and elsewhere [3].The paper can be considered as an introduction to the subject of quantization from Connes’point of view and thus serves too a pedagogical purpose.Rieffel’s requirements are stronger that those of formal deformation theory,ex-tant for any Poisson manifold[4].To our mind,however,the fact that T∗M,for M Riemannian,possesses a strict quantization,was indeed proved by Lands-man in the path-breaking paper[5].Nevertheless,the noncommutative geo-metry approach presents several advantages,not least that the C∗-theoretical aspects come in naturally.The procedure was suggested by Landsman himself, even before[2]was in print,at the end of his paper.For the sake of simplicity, we deal here with the nonequivariant case only.The plan of the article is as follows.In thisfirst Section,after introducing groupoids and the tangent groupoid construction,we give an elementary dis-cussion of Connes’recipe for quantization.Wefind the intersection,for the case M=R n,of the family of what could be termed“Connes’quantization rules”with the ordinary quantum-mechanical ordering prescriptions.Groupoids are here regarded set-theoretically,questions of smoothness being deferred to Sec-tion2.We show that Moyal’s quantization rule belongs to the collection of Connes’quantization rules and in fact is singled out by natural conditions in strict deformation theory.In Section2we spell out our variant of Connes’tangent groupoid construction in full detail.The heart of the matter is the continuity of the groupoid product, for which we give two different proofs.In Section3,by means of the mathematical apparatus of Section2,we restate Landsman’s partial answer to Rieffel’s question.In particular,we rework the existence proof for a strict quantization of the Moyal type(in the sense of being both real and tracial)on Riemannian manifolds.Moreover,using a strong form of the tubular neighbourhood theorem,we show that there exists what Rieffel calls aflabby quantization[1].The paper concludes with a discussion on the C∗-algebraic aspect of the tangent groupoid construction and its relation to the index theorem.1.2Basic facts on groupoidsThe most economical way to think of a groupoid is as a pair of sets G0⊂G, and to regard elements of G as arrows and elements of G0as nodes.Definition1A groupoid G⇉G0is a small category in which every mor-phism has an inverse.Its set of objects is G0,its set of morphisms is G.A group is of course a groupoid with a single object.The gist of the definition is conveyed by the following example.1.2.0.1Partial isometries Consider a complex Hilbert space H.The col-lection of unitary arrows between closed subspaces of H obviously defines a groupoid,for which G is the set of partial isometries{w∈L(H):ww∗w=w} and G0is the set of orthogonal projectors in L(H).Recall that,given w,we can writeH=ker w⊕im w∗=ker w∗⊕im w.Hence w∗w is the orthogonal projector with range im w∗,while ww∗is the orthogonal projector with range im w;of course,w∗is the inverse of w.We naturally identify two maps r,s(respectively“range”and“source”)from G to G0:r(w):=ww∗and s(w):=w∗w.Also,it is natural to considerG(2):={(u,v)∈G×G:u∗u=vv∗}.This defining equation is a sufficient condition for the operator product uv to be a partial isometry,as follows from the simple calculationuv(uv)∗uv=uvv∗u∗uv=uvv∗v=uv.The example motivates a more cumbersome restatement,that includes all the practical elements of the definition.Definition2A groupoid G⇉G0consists of:a set G,a set G0of“units”with an inclusion G0֒→G,two maps r,s:G→G0,and a composition law G(2)→G with domainG(2):={(g,h):s(g)=r(h)}⊆G×G,subject to the following rules:(1)if g∈G0then r(g)=s(g)=g;(2)r(g)g=g=gs(g);(3)each g∈G has an“inverse”g−1,satisfying gg−1=r(g)and g−1g=s(g);(4)r(gh)=r(g)and s(gh)=s(h)if(g,h)∈G(2);(5)(gh)k=g(hk)if(g,h)∈G(2)and(gh,k)∈G(2).The examples of groupoids we shall use have a differential geometricflavour instead.1.2.0.2A vector bundle Eπ−→M Here G=E is the total space,G0= M is the base space,r=s=πso that G(2)= x∈M E x×E x(the total space of the Whitney sum E⊕E),and the composition law isfibrewise addition.1.2.0.3The double groupoid of a set Given a set M,take G=M×M and G0=M,included in M×M as the diagonal subset∆(M):={(x,x): x∈M}.Define r(x,y):=x,s(x,y):=y.Then(x,y)−1=(y,x)and the composition law is(x,y)·(y,z)=(x,z).We shall generally call G0the diagonal of G.1.3Connes’tangent groupoidsWe recall some basic concepts of differential geometry,particularly sprays and normal bundles,that we shall later use.Given a symmetric linear connection on a differentiable manifold M,one can define a vectorfieldΓon T M whose value at v∈T M is its horizontal lift to T v(T M).This vectorfield is called the geodesic spray of the connection and its integral curves are just the natural lift of geodesics in M.The geodesic spray is a second-order differential equation vectorfield satisfying an addi-tional condition of degree-one homogeneity which corresponds to the affine reparametrization property of geodesics.More generally,a second-order differ-ential equation vectorfield is said to be a spray if the set of its integral curves is invariant under any affine reparametrization:these curves are called geodesics of the spray.Given a sprayΓ,there is a symmetric connection whose geodesic spray isΓ;and conversely,the connection is fully determined by its geodesic spray,that can also be used to construct the exponential map exp:T x M→M.A Riemannian structure on M determines one symmetric linear connection, the Levi-Civita connection,and consequently a Riemannian spray.Now let Y0be a submanifold of a manifold Y.The normal bundle N YY0toY0in Y is defined as the vector bundle N YY0:=T Y/T Y0,where the notationmeans that its base is Y0and itsfibre is given by the equivalence classes of the elements of the tangent bundle T Y under the relation:X1∼X2,for X1,X2∈T q Y with q∈Y0,if and only if X1=X2+V for some V∈T q Y0.The usual way to work with such a structure is to choose a representative in each class, thereby forming a complementary bundle to T Y0in T Y restricted to Y0.There is no canonical choice for the latter,in general.When a Riemannian metric is provided on Y,there is a natural definition of the complementary bundle as the orthogonal complement of the tangent space T q Y0in T q Y(although other choices may be convenient,even in the Riemannian case).Once we have chosena suitable representative of each class,the bundle N YY0becomes a subbundleof T Y and we can consider the exponential map exp of T Y restricted to N YY0.We recall also that a tubular neighbourhood of Y0in Y is a vector bundle E→Y0,an open neighbourhood Z of its zero section and a diffeomorphism of Z onto an open set(the tube)U⊂Y containing Y0,which restricts over the zero section to the inclusion of Y0in Y[6].We say that the tubular neighbourhood is total when Z=E.The main theorem in this context establishes that,given a spray on Y,one can always construct a tubular neighbourhood by making use of the corresponding exponential map.When the spray is associated to a Riemannian metric,one can always have a total tubular neighbourhood, because a Euclidean bundle is compressible,i.e.,isomorphic as afibre bundle to an open neighbourhood of the zero section:see[6].For suitable 0>0,one can then define the normal cone deformation(a sort of blowup in the differentiable category)of the pair(Y,Y0),denotedM YY0,by gluing together Y×(0, 0]with N YY0as a boundary with the help ofthe tubular neighbourhood construction[7].The construction is particularly interesting when(Y,Y0)is a groupoid,in that it gives a“normal groupoid”with diagonal Y0×[0, 0].We consider a particular case of this construction,the tangent groupoid.Let M now be an orientable Riemannian manifold.Denote by N∆,rather thanN M×MM ,the normal bundle associated to the diagonal embedding∆:M→M×M.We can identify T M⊕T M with the restriction of T(M×M)to ∆(M),and the tangent bundle over M is identified to{(∆(q),X q,X q):(q,X q)∈T M};thereby the normal bundle N∆to M in M×M can a priori be identified with{(∆(q),ϕ1q X q,ϕ2q X q):(q,X q)∈T M},(1) whereϕ1,ϕ2∈End(T M)are any two bundle endomorphisms(i.e.,continuous vector bundle maps from T M into itself)such that the linear mapϕ1q−ϕ2q on T q M is invertible for all q∈M;this we write asϕ1−ϕ2∈GL(T M).We shall assume,for definiteness,that eachϕ1q−ϕ2q is homotopic to the iden-tity:in other words,that there is an isomorphism of oriented vector bundles between T M and N∆.The tangent groupoid G M⇉G0M,according to Connes,is essentially thenormal groupoid M M×MM modulo that isomorphism[2].That is to say,wethink of the disjoint union G M=G1⊎G2of two groupoidsG1:=M×M×(0, 0],G2:=T M,where G1is the disjoint union of copies M×M×{ }of the double groupoidof M,parametrized by0< ≤ 0.The compositions are(x,y, )·(y,z, )=(x,z, ) (q,X1)·(q,X2)=(q,X1+X2)with0< ≤ 0, if X1,X2∈T q M.The topology on G M is such that,if(x n,y n, n)is a sequence of elements of G1with n↓0,then it converges to a tangent vector(q,X)iffx n→q,y n→qandx n−y nLet a(q,p)be a function on T∗R n.Its inverse Fourier transform in the second variable gives us a function on T R n:F−1a(q,X)= R n e i Xp a(q,p)d p.To the function a,Connes’prescription associates then the following family of kernels:k a(x,y; ):=F−1a(Φ−1 (x,y, )).The map a→k a is thus linear.We get the dequantization rule by Fourier inversion:a(q,p)=1∂q +∂∂yK=0,(4)i.e.,K depends only on the combinations x−y and q−1satisfying(4)and(5)is of the formK(q,p,x,y; )=12(x+y))/ )dθ,where f should have no zeros,on account of invertibility;that can be rewritten asK(q,p,x,y; )=1(2π )nR nδ(x−q− ϕ1X)δ(y−x+ X)exp(−i pX)d X.This integral can be rewritten as(2π )−nδ(x−q−ϕ1(x−y))e−i p(x−y)/ , and further transformed into12(x+y)−q+(1which is of the previously given form,withf (θ,τ)=exp(i θ(1(2π )nR n δ(x −q − ϕ1q X )δ(y −x + (ϕ1q −ϕ2q )X )exp(−i pX )d X,we would have had two difficulties:equivariance under translation of the posi-tion would be lost since ϕ1q =ϕ1q ′in general,and equivariance under transla-tion of the momentum would be lost since ϕ1q −ϕ2q =id T q R n in general;this bears on the physical meaning of Connes’limit condition (2).It would be tempting,also in view of (2),to add the time variable to the mathematical apparatus of this paper,in the spirit of Feynman’s formalism,and to study the interchangeability of the limits t ↓0and ↓0.Happily,Moyal’s rule is included among Connes’rules:it follows from themost natural choice ϕ1=12(see the discussion in [12]).We arrive at the following conclusion.Proposition 3Moyal quantization rule is the only real quantization of the Connes type,in the case M =R n .Connes’own choice in [2]is ϕ1=0,ϕ2=−1;this corresponds to the “standard”ordering prescription,in which the quantization of q n p m is Q n P m ;whereas the choice ϕ1=1,ϕ2=0leads to the “antistandard”ordering,in which the quantization of p m q n is P m Q n .Note that all of Connes’prescriptions are tracial.In order to obtain the “normal”and “antinormal”prescriptions of use in field theory,which are real but not tracial,one would have to complexify Connes’construction;we shall not go into that.Following Landsman,we will strengthen the definition of strict quantization in order to remain in the real context (see Section 3).Our elementary discussion in this subsection leads to presume that real strict quantization can be done in the framework of tangent groupoids;this is tantamount to the generalization of the Moyal rule to arbitrary manifolds endowed with sprays.All the pertinent differential geometric constructions for that purpose are taken up in the next section.2The tangent groupoid construction:the Moyal versionDefinition4A smooth groupoid is a groupoid(G,G0)together with differ-entiable structures on G and G0such that the maps r and s are submersions, and the inclusion map G0֒→G is smooth as well as the product G(2)→G.Note the dimension count:if dim G=n,dim G0=m,then dim G(2)=2n−m.The definition of smooth groupoid is taken from[2].Now we establish his Proposition II.5.4,there left unproven by Connes.Proposition5The tangent groupoid G M to a smooth manifold M is a smooth groupoid.2.1G M as a manifold with boundaryThe groupoid G1=M×M×(0, 0],that will be the interior of G M(plus a trivial“outer”boundary,which we neglect to mention in our following argu-ments)is given the usual product manifold structure.To complete the defi-nition of the manifold structure of G,consider the isomorphism from T M toN∆given by(q,X q)→(q,12X q).Consider also the product manifoldT M×[0, 0].We choose a spray on M—this is provided,for instance,by a choice of Riemannian metric on M—and define a mapT M×[0, 0]⊃UΦ−→G M,where U is open in T M×[0, 0]and includes T M×{0},as follows:Φ(q,X q, ):=(exp q( X q/2),exp q(− X q/2), )for >0,andΦ(q,X q,0):=(q,X q)for =0.Here exp denotes the exponential map associated to the spray:we know that, for afixed ,the exponential map defines a diffeomorphism of an open neigh-bourhood V of the zero section of T M onto an open neighbourhood of the diagonal in M×M;and we decide that a point(q,X q, )is in U ifΦ(q,X q, ) is contained in V.Therefore,both the existence and,for a suitable choice of U,the bijectivity of the mapΦfollow from the tubular neighbourhood theorem.As U is anopen(sub)manifold with boundary,we can carry the structure of manifold with boundary to G M,obtaining that T M is the boundary of the groupoid G M in the topology associated to that structure.The diagonal is obviously M×[0, 0].We remind the reader that Connes uses instead the chart given by(q,X q, )→(q,exp q(− X q), )for >0,and(q,X q,0)→(q,X q)for =0.From now on,a particular Riemannian structure is assumed chosen,and when convenient we shall also assume,as we then may,that U is the whole of T M×[0, 0].To continue the proof of smoothness of G M,we need to check that the various mappings have the required smoothness properties.The basic idea is to pull all maps on G M back to T M×[0, 0]and prove smoothness there.Considerfirst the inclusion map i:G0M→G M.It is obvious that the restriction of i to M×(0, 0]is smooth in its domain.If we now consider its restriction to i−1(Φ(U))and compose it withΦ−1,we obtain a map which can be written as:Φ−1◦i(x, )=Φ−1(x,x, )=(x,0x, )for >0,andΦ−1◦i(x,0)=Φ−1(x,0x,0)=(x,0x,0)for =0.This map is obviously smooth in its domain and,asΦis a diffeomorphism, i is smooth in its domain.We shall consider now the range and source maps.The smoothness of both maps when restricted to M×M×(0, ]is again obvious,as is the fact that they are of maximum rank(hence submersions when restricted to this domain).The composition ofΦwith the restriction of r toΦ(U)is expressed as: r◦Φ(q,X q, )=r(exp q( X q/2),exp p(− X p/2), )=(exp q( X q/2), )for >0andr◦Φ(q,X q,0)=(q,0)for =0.Again this map is smooth and of maximum rank in its domain,so that r must also be a submersion.The corresponding proof for the source map is analogous.2.2The geometrical structure of G(2)M⊂G M×G MTo define a differentiable structure for G(2)M,we proceed as in the previous case,by defining a bijection between an open set in G(2)M and an open set in a manifold with boundary,and transporting the differential structure via the bijection.What kind of manifold is G(2)M?The product has to be a smooth mapping between two manifolds with boundary,mapping the boundary on the bound-ary and the interior on the interior.Thus,from the definition of the product it is clear that the boundary of the manifold should be the Whitney sum T M⊕T M,arising as the pullback with respect to the diagonal injection of M in M×M of the product bundle T M×T M.Points in the interior of the manifold are pairs of the form:((x,y, ),(y,z, )). Therefore,we need to define a differential structure in G(2)M in such a way that it becomes a manifold with boundary T M⊕T M,its interior being diffeomorphic to M×M×M×(0, 0].Consider now T M×T M×[0, 0].This is a manifold with boundary,with a natural differential structure.Let(q′,q,X q′,Y q, )be a point in this mani-fold;we use our construction for G M on each T M separately,i.e.,we set the bijection:(q′,q,X q′,Y q, )↔{(e X q′/2q′,e− X q′/2q′, ),(e Y q/2q,e− Y q/2q, )}for >0—with an obvious notation for exp.Those points that interest usresult just from imposing that e− X q′q′=e Y q q.We shall see that this constraintdefines a regular submanifold of T M×T M×[0, 0].For that,for afixed value of ,consider the sequence of mapsΨ:T M×T MΦ−→M×M×M×Mπ23−→M×MΦ−1−→T M→R n given by(q′,q,X q′,Y q)→(e X q′/2q′,e− X q′/2q′,e Y q/2q,e− Y q/2q)→(e− X q′/2q′,e Y q/2q)→(r,X r)→X r,where the(r,X r),that depend on ,are found soΦ(r,X r)=(e− X q′/2q′,e Y q/2q).This composition defines a differentiable mapping of constant rank(it is com-posed of two bijections and two projections onto factors of a product).We then extendΨto a map from T M×T M×[0, 0]to R n×[0, 0]and have then thatΨ−1({0}×[0, 0])defines a regular submanifold S of T M×T M×[0, 0], whose differentiable structure we use to define the structure of manifold with boundary on G(2).[It is perhaps not entirely clear that the boundary of S is T M⊕T M,as we wish.But remember that r depends on through the above manipulations.If we have a sequence(q′n,q n,X q′n ,Y qn, n)in S,with n↓0,its limit is a point(s,s,X,Y)in T M×T M where s= lim q′n=lim q n=lim r( n).]2.3Continuity of the productA very simple argument with Riemannianflavour allows one to prove at least continuity of the product operation in the tangent groupoid.Consider a se-quence in G(2):{(x n,y n, n),(y n,z n, n)}=Φ(q′n,q n,X n,Y n, n)with limit on the boundary.We need to check that the limit of the products coincides with the product of the limits in G M.For elements close enough to the boundary,we know that(s(n),Z s(n))exists,sox n=e Z s(n)/2s(n),z n=e− Z s(n)/2s(n).Assume thatq′n→s,q n→s,X n→A,Y n→B as ↓0. Now,we have(e Z s(n)/2s(n),e− Z s(n)/2s(n))=(e X n/2q′n,e− Y n/2q n)→(s,s)as ↓0,hencelim s(n)=s.It remains to show that lim Z s(n)=A+B.This will follow if we prove Z n= X n+Y n+o( n).But that follows from consideration of the small triangle with vertices x n,y n,z n,formed by geodesics through q′n,q n,s(n)with directions ±X n,±Y n,±Z s(n).One sees that2(X n+Y n−Z s(n))is approximately a circuit around this triangle; by the Gauss–Bonnet theorem,we conclude thatX n+Y n−Z s(n)≈O( 2).2.4A functorial proof of smoothnessLemma6Let there be given two closed submanifolds X0֒→X and Y0֒→Y and a smooth mapping f:X→Y which satisfies f(X0)⊂Y0.Then theinduced mapping˜f:M XX0→M YY0between the corresponding normal conedeformations is also smooth.This is Lemma2.1in[13],where no proof is offered.We give some details of the lemma and then of its application.First of all,if f is a smooth mapping from X to Y such that f(X0)⊂Y0,the image under f∗of the tangent bundle to X0is a subbundle of T Y0,implying the existence of an induced mapping between the respective normal bundles,that we shall continue to call f∗.Now we define˜f by˜f(x, ):=(f(x), )for >0,and˜f(a,Xa):=(f(a),f∗(X a))for(a,X a)an element of the normal bundle and =0.Now,differentiability of˜f follows from the limitlim ↓0 −1exp−1f(a)[f(exp a( X a))]=f∗(X a).The application to the groupoid operations in our context is plain.We discuss the product operation,the only relatively tricky one.Let m:G(2)→G be the groupoid multiplication.Now G(2)∩(G0×G0)=∆(G0)and m(G(2)∩(G0×G0))=G0;indeed,if(u,v)∈G(2)∩(G0×G0)then u=s(u)=r(v)=v by property(i)of Definition2,and so uv=us(u)=u by property(ii).Therefore˜m:M G(2)G0→M GG0is smooth.It remains to prove that M G(2)G0is diffeomorphicto(M GG0)(2).But this is clear on examining the definitions:indeed,M G(2)G0=G(2)×(0, 0]⊎N G(2)G0={(g,h, ):s(g)=r(h)}⊎{(u,X u,Y u):X u,Y u∈N G G0},whereas(M G G0)(2)={(g, 1;h, 2):s(g)=r(h), 1= 2}⊎{(u,X u;v,Y v):u=v}.For G=G M,that boils down to T M⊕T M≈N M×M×MM .Notefinally thedimension count:the dimension of(M GG0)(2)is2(dim G+1)−dim(G0+1)=2dim G−dim G0+1,which is clearly the same as the dimension of M G(2)G0. This completes the proof of Proposition5.3Tangent groupoids and strict quantization3.1The deformation conditionsIn our definition of quantization,we actually strengthen some of Rieffel’s re-quirements.Regard T∗M as a Poisson manifold and consider the classical C∗-algebra A0:=C0(T∗M)of continuous functions vanishing at infinity.We choose a dense subalgebra A0(there is considerable freedom in that,but,to fix ideas,we think of the functions whose Fourier transform in the second argument has compact support),and we search for a family of mappings Q into noncommutative C∗-algebras A such that the following relations hold for arbitrary functions in A0:(1)the map → Q (f) is continuous on[0,h0)with Q0=I;(2)lim →0 Q (f1)Q (f2)−Q (f2)Q (f1)−i Q ({f1,f2}) =0;Those are Rieffel’s strict quantization conditions(Question23of[1]),to which we add:(3)the asymptotic morphism condition lim →0 Q (f1)Q (f2)−Q (f1f2) =0;(4)the reality condition Q (f∗)=Q (f)∗;and also(5)the traciality condition Tr[Q (f1)Q (f2)]= T∗M f1(q,p)f2(q,p)dµ (q,p);where we use the same symbols∗(a bit overworked,admittedly)and · for the adjoint and norm in every C∗-algebra.Axioms(1)to(4)are a slight variant of Landsman’s axioms.That the tangent groupoid construction provides an answer to the twentieth query by Rieffel fol-lows indeed from Landsman’s calculations in[5].There is no point in repeatingthem here,and we limit ourselves to the necessary remarks tofit them in the tangent groupoid framework.Axiom(5)is employed to further select a unique recipe.In[1],Rieffel introduces the important concept of“flabbiness”:a deformation quantization isflabby if it contains the algebra of smooth functions of compact support on M.The constructions performed in this paper require in princi-ple only the existence of sprays,in order to use the tubular neighbourhood theorem.However,in that caseflabbiness is not guaranteed(see below).3.2The C∗-algebra of a groupoidThe natural operation on functions of a groupoid is convolution:(a∗b)(g):= {hk=g}a(h)b(k)= {h:r(h)=r(g)}a(h)b(h−1g)but for this to make sense we need a measure to integrate with.We can either define a family of measures on thefibres of the map r,G x:={g∈G: r(g)=x}for x∈G0(see the detailed treatments given by Kastler[14]and Renault[15])or we canfinesse the issue by ensuring that the integrand is always a1-density on each G x.In the second approach,one uses half-densities,rather than functions.We summarize it here,for completeness.Denote the typicalfibre of s by G y:= {g∈G:s(g)=y}for y∈G0.Since r and s are submersions,thefibres G x and G y are submanifolds of G of the same dimension,say k.If x=r(g)and y=s(g),thenΛk T g G x andΛk T g G y are lines.LetΩ1/2g be the set of maps ρ:Λk T g G x⊗Λk T g G y→C such thatρ(tα)=|t|1/2ρ(α)for t∈R.This is a(complex)line,and it forms thefibre at g of a line bundleΩ1/2→G, called the“half-density bundle”.Let C∞c(G,Ω1/2)be the space of smooth, compactly supported sections of this bundle.For a,b in this space,the con-volution formula makes sense and a∗b∈C∞c(G,Ω1/2)also.The C∗-algebra of the smooth groupoid G⇉G0is the algebra C∗(G)obtained by completing C∞c(G,Ω1/2)in the norm a :=sup y∈G0 πy(a) ,whereπy is the representa-tion of C∞c(G,Ω1/2)on the Hilbert space L2(G y,Ω1/2)of half-densities on the s-fibre G y:πy(a)ξ:g→ G y a(h)ξ(h−1g)where one notices that the integrand is a1-density on G y.If G=M×M,we get just the convolution of kernels:(a∗b)(x,z):= M a(x,y)b(y,z)d ywhere d y denotes integration of a1-density on M parametrized by y.This business of half-densities is very canonical and independent of preassigned measures.However,in our case,if M is an oriented Riemannian manifold,we may use the volume form dν(x):=g(q)d Ywhere we may take f1(x,·)and f2(x,·)in C∞c(T x M).The Fourier transformF a(q,p)=1g(q)d Xreplaces convolution by the ordinary product on the total space T∗M of the cotangent bundle.This extends to the isomorphism also called F:C∗(T M)→C0(T∗M),with inverse:F−1b(q,X)= T∗q M e i pX b(q,p)d n p g(q).3.3The quantization and dequantization recipesLetγq,X be the geodesic on M starting at q with velocity X,with an affine parameter s,i.e.,γq,tX(s)≡γq,X(ts).Locally,we may writex:=γq,X(s), y:=γq,X(−s),with Jacobian matrix∂(x,y)Then one has the change of variables formula:M×MF(x,y)dν(x)dν(y)= M T q M F(γq,X(12))J(q,X;1g(q)d X dν(q), where we introduceJ(q,X;s):=s−n det g(γq,X(−s))∂(q,X)(s).This object can be computed from the equations of geodesic deviation[5].The crucial estimate isJ(q,X,1∂qν;the second one using the variation of the tangent vector,˜hµν(q,X,s)=∂xµ1(q,X,s)Here J−1/2,J1/2mean the corresponding multiplication operators.It is pos-sible to rewrite the formulae so J only appears once,but then we would lose property(5).So there is actually a family of“Moyal”quantizations in the general case,unless we demand traciality(as we do).This is a feature of the nonflat case:note that J=1in the framework of our Section1.In the main development in[5]J actually only appears in the dequantization formula,but again Landsman gives indication on how to modify the formulae to get tracial-ity.Those factors are analogous to the preexponential factors that appear in the semiclassical expression for the path integral:see,e.g.,[16,p.95].One can insert them in several ways without altering the axioms,precisely on account of(6).A bit more explicitly,the previous formulae arek a(x,y; )=J−1/2(q,X,1)k a(x,y; )],(7)2where(x,y)and(q,X)are related by x=γq,X(1).2The long but relatively straightforward verifications in[5]then show that we have defined an(obviously real)preasymptotic morphism which moreover is a strict quantization from C∞c(T∗M)to K(L2(M)).In addition,the tracial property(5)is satisfied.In the Riemannian context we have total tubular neighbourhoods,and it is clear that Fourier transforms of smooth functions of compact support in T∗M decay fast enough that our formulae make sense for them;therefore the quantization isflabby.We make afinal comment on the uniqueness,or lack of it,of the quantiza-tion considered here.Among our choices are those we make to establish the isomorphism between T M and N∆on which the whole construction hinges. Regarding only the differential structure,those choices are parametrized by, say,the pair(ϕ1+ϕ2,ϕ1−ϕ2)∈End(T M)×GL(T M).However,the identifi-cation of the normal bundle with the orthogonal bundle to the diagonal leads to the equationϕ1+ϕ2=0,if we take the natural metric on M×M.To the same equation leads the reality constraint(4),as a simple calculation from(7) shows.After so narrowing the freedom of parameter choice to GL(T M),semitraciality implies at least det(ϕ1−ϕ2)=1.Connes’condition(2)and our discussion in Section1strongly suggest to adopt the restrictionϕ1−ϕ2=1.Let us agree to call all the quantizations(parametrized by End(T M))for which the last equation holds quantizations of the Connes type.Then we have proved: Theorem7For any Riemannian manifold,the only real quantization rules of the Connes type are Moyal quantizations.。
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量化金融标准委员会Standard Committee of Quantitative Finance量化金融分析师（AQF®）全国统一考试模拟题说明：本场考试中的代码都应采用Python 3.X版本作答。
1.单选题（每题1分，本部分共20分）:只有一个正确答案，选对得1分，选错或不选得0分。
1.1 技术分析是重要的投资分析方法之一。
其中，起源于日本德川幕府时代的“K线图”（又称蜡烛图、阴阳线）是常用的技术分析方法。
当我们发现某交易日的K线为无下影线阴线时，那么该K线实体的上边线表示（）？A. 最高价B. 收盘价C. 最低价D. 开盘价1.2 以下关于各大量化投资交易策略的描述中，不正确的是（）？A. 在多因子策略中，一般而言，所选因子的相关性越低越好B. 一般而言，资金流对个股的短期波动影响更大C. 动量反转策略利用的是价格均值回归的特性D. 趋势跟踪策略本质上是一种追涨杀跌的策略，因此并不具有任何盈利的可能性1.3 李明，AQF，某量化基金经理，他在量化投资交易的过程中，发现回测收益往往会大幅高于实盘收益，研究发现造成这种现象的原因有很多，其中，常见的一种原因是在回测的过程中使用了未来数据，未来数据会使得回测收益虚高。
那么在下列量化策略研究过程中，哪个选项最有可能没有使用到未来数据（）？A. 在策略回测的过程中，采用某天的最低价作为当天的买入成本价B. 使用整个样本数据对策略参数进行寻优后，使用该优化后的参数进行策略回测，并对该策略进行有效性评估C. 以发出交易信号后的下一天的开盘价作为策略的成交价，未设置滑点D. 以当前沪深300成分股为研究对象，研究过去10年沪深300成分股的选股策略1.4 李明，AQF，某量化基金经理，正在研究事件驱动型套利策略，以下哪个选项描述了事件驱动型套利策略（）？A. 基金经理持有目前或者预期会发生诸如以下交易事项的公司的金融产品：（包括但不限于）合并、重组、财务危机、股权收购、股东回购、发行债务交换、证券发行或其他资本结构调整B. 基金经理根据潜在的宏观经济变量及其对股票、固定资产、货币和大宗商品市场的影响，进行相关交易C. 基金经理基于多个证券估值差异及其关系的理论进行交易D. 基金经理在现货市场和衍生品市场进行方向相反的操作1.5 字符串格式化是量化投资策略编写过程中常用的方法，那么以下哪种代码可以用来实现浮点数格式化（）？A．%cB. %dC. %fD. %s1.6 李明，AQF，某量化基金经理，在1.2308做空欧元/美元的差价合约，之后欧元/美元汇率跌至1.2133，李明可以通过以下哪个类型的委托单来实现继续持有空头仓位的同时控制回撤风险（）？A. 市价买单B. 市价卖单C. 限价买单D. 止损买单1.7 当横线处填入（）时，代码打印输出的结果是列表中所有的深交所上市的股票代码？（注意：上交所代码以6开头，深交所代码以0或3开头）for stock_code in ['002003', '600015', '300001', '002300']: if stock_code.startswith('6'):_____print(stock_code)A．raiseB. continueC. passD. break1.8 李明，AQF，某量化基金经理，他在进行策略研究时需要从DataFrame数据类型stock_base_data中提取2018-01-03至2018-01-05（含01-05）时间段中的股票PE、CLOSE数据，该DataFrame如下：则李明提取数据时可以使用的代码为（）？A. stock_base_data.iloc['2018-01-03':'2018-01-05', ['PE', 'CLOSE']]B. stock_base_data.loc['2018-01-03':'2018-01-05', ['PE', 'CLOSE']]C. stock_base_data.loc[['PE', 'CLOSE'], '2018-01-03':'2018-01-05']D. stock_base_data.iloc[2:4, ['PE', 'CLOSE']]1.9 李明，AQF，某量化基金经理，想要评估长期持有的某只股票在过去三天的总体表现。
遗传名词解释13Malformation ：is a primary structural defect of an organ or part of an organ which results from an intrinsic abnormality in development.Disruption: refers to abnormal structure of an organ or tissue as result of external factors disrupting the normal developmental processDeformation is a defect which results from an abnormal mechanical force which distorts an otherwise normal structure.Sequence–findings occur as a consequence of a cascade of events initiated by a single primary factor, e.g., Potter sequenceSyndrome–consistent and recognizable patterns of abnormalities with a known cause, e.g., Down syndromeAssociation–Non-random clustering of certain malformations that cannot be explained on the basis of a sequence or syndrome, e.g., VA TER12Gene therapy：treatment of human disease by transferring genetic material (DNA or RNA) into the patient.In vivo gene therapy: genetic material transferred directly into cells in a patient.Ex vivo gene therapy: cells removed from a patient with genetic material inserted in vitro, prior to transplanting the modified cells back into the patient.RNA interference (RNAi)：A way to ‘silence’genes by preventing the formation of proteins that they code for, which has taken advantage of an intermediate step between DNA and protein.11Genome:Whole DNA sequences of an organism.genetic map：the relative position of genes and markers along a chromesome ,as determined by genetic linkage information.physical map :the actual location of the genes along the chromosome have been determined by direct physical techniques ,such as restriction mapping and physical cloning…….contigChromosome walking –construction of an ordered assembly of clones extending from a start point.Phylogenetic analysis:Deduction of evolutionary relationships through sequence comparison10Genetic mapping: relative position of genes along a chromosome as determined by genetic linkage information.Functional cloning:Identification of a disease gene through knowledge of its protein productLinkage analysis: To locate the disease gene through analysis of co-transmission (co-segregation) of genetic markers.Association study: correlated occurrence of specific alleles at a genetic locus and the diseasein population (candidate genes) .HTF/CpG islands:Clusters of undermethylated CpG dinucleotides have been found near the transcription initiation sites of the 5’end of many genes, so called methylation-free HpaII tiny fragments (HTF), or CpG islands.Expressed sequence tags(ESTs):ESTs are sequence-specific primers from cDNA clones designed to identify sequences of expressed genes. Provide a rich source of partial sequences for a large number of genes.……….homologs9Somatic mutations: only found in tumor but not in the normal cells of an individual. Germline mutation: presents in all cells of an individual and may be passed on to subsequent generations.An oncogene is an altered version of a proto-oncogene (resulting from mutation, over-expression, or amplification) that contributes to neoplastic transformation. Oncogenes are dominant-type cancer transforming genes.LOH: loss of one allele in a tumor cell from a chromosomal region for which the individual’s normal cells are heterozygous.8Karyotype：Total sum of chromosome, sex chromosome .Normal male:46, XY ;Normal female: 46, XXAneuploidy (非整倍体): extra copy or absence of single chromosomes. trisomy (三体) (e.g. trisomy 21 in Down syndrome) . monosomy (单体) (e.g.45,X in Turner syndrome) nondisjunction (不分离): failure of chromosomes to separate normally during cell division.mitotic mosaic (嵌合体): an individual with two or more populations of chromosomallydifferent cells;meiotic(ⅠorⅡ) trisomic zygotesmonosomicGenomic Imprinting (基因组印刻):Differential expression of genetic information depending on whether it was inherited from the father or the mother.SRY(sex-determining region on Y) contains: highly conserved DNA-binding domain; acts as: testis-determining factorFragile X Syndrome (a form of X-linked Mental Retardation)Karyotype: 46,fra(X)Y Frequency:1/1250 males;1/2000 females.Features: narrow face, prominent forehead, jaw and ears, large testes, moderate to sever mental retardation; affected females milderDynamic mutation: consists of triplet repeat sequences which, in affected person, occur in increased copy number when compared to the general population.7Inborn errors of metabolism: a genetically determined biochemical disorder in which a specific enzyme defect produces a metabolic block that may have pathological consequences. 6Gene cluster:any group of two or more closely linked genes that encode for the same or similar products.Gene family: a set of genes in one genome all descended from the same ancestral gene.5vector :After a specific fragment of DNA is cut with RE, it can be linked by DNA ligase with a DNA segment capable of autonomous replication. It is carrier of recombinant DNA or RNA, such is called a vector (replicon).Hybridization is the ability of a single-stranded DNA or RNA to anneal by base-pairing to its complementary single strand, while failing to anneal to an unrelated sequence.………..DNA liabraryRFLP (restriction fragment length polymorphism): DNA sequence variation results in the gain or loss of a restriction site.VNTR (variable number of tandem repeats) refers to a locus that is hypervariable because of tandemly repeated DNA sequences. Presumably variability is generated by unequal crossing over or slippage during replication.…………SNPGenotyping: research the genetic constitution of an individual or,more specifically,the alleles at specific loci.5Susceptibility：The risks of genetically determining to diseases for individual (cumulative effect of genes) .Liability:The probabilities of developing a disease which depend on the interaction of various genetic and environmental factors.Candidate genes: genes that encode proteins known or thought to be involved in the disease process.……………identical-by-descent (IBD)4Consanguinity: couples sharing at least one common ancestor no more remote than great-great-grandparentBiological fitness (f): measure of fertility f = 0 ~ 1 (0: genetic lethal, 1: normal allele) Gene Flow:gradual diffusion of genes from one population to another ,as a result of migration and intermarriage.founder effect: the high frequency of a mutant gene in a rapidly expanding population founded by a small ancestral group in which one or more of the founders was ,by chance ,a carrier of the mutant gene3Proband（先证者）:the first affected family member coming to medical attention. Alleles （等位基因）are alternative forms of a gene at a given locus.Compound heterozygote（复合杂合子）: An individual with two different mutant alleles at a given locus.Mendelian disease: refers to diseases that are the result of a single mutant gene that has a large affect on phenotype and that are inherited in simple patterns similar to or identical with those described by Mendel for certain discrete. It include Autosomal Dominant Inheritance, Autosomal Recessive Inheritance, X-Linked Recessive Inheritance, X-Linked Dominant InheritanceExpressivity It refer to the nature and severity of the phenotype.Penetrance（外显率）: all-or-none phenomenon that refers to the clinical expression or lack ofit, of the mutant gene.2karyotype.Each species has a characteristic number and size of chromosomes, known as the karyotype Human karyotype: 46, XX (XY)Introns An intervening or non-coding sequence of DNA that separates the protein-coding segments of eukaryotic genes into discrete blocks (exons).exons A segment of a eukaryotic gene presented in the mature mRNA molecule after the primary transcript is spliced to remove introns. Exons contain all of the protein-coding sequences of a gene as well as the untranslated sequences (UTR) located at the 5’and 3’ends of the eukaryotic mRNA.promotor Promoter is a regulatory sequence located immediately 5’to the gene that interact with RNA polymerase and other components of the transcription machinery to initiate transcription.enhancer Enhancers, e.g., GC box (~80 bp) and CAAT box (further upstream) , are a regulatory DNA sequence that can increase the transcription of a nearby gene or over a considerable distance even if inverted. It controls the abundance, tissue specificity, and temporal pattern of transcription of an associated gene.Translation .the process by which a polypeptide is synthesized upon a ribosome under the direction of the coded instructions provided by an mRNA moleculeTranscription the process by which a complementary RNA molecule is synthesized upon a DNA template by the action of RNA polymeraseMissense mutation: a single DNA base substitution resulting in a codon specifying a different amino acid.Nonsense mutation: a single DNA base substitution resulting in a stop codon.Frameshift mutation:a mutation involving a deletion on insertion that is not an exact multiple of 3, which changes the reading frame of the gene from that point onward, leading to a completely abnormal carboxyl terminus of the protein.Synonymous (silent) mutations: a single DNA base substitution resulting in no alteration of amino acid.Non-synonymous mutations: including missense, nonsense and frameshifing mutations1Crossing over: the exchange of chromosomal material between homologous chromosomes by breakage and reunion.Linkage: Condition in which two and more non-allelic genes tend to be inherited together. somatic cell genetic disorders In contrast to the above three categories in which the genetics abnormality is found in the DNA of all cells in the body including germ cells (sperms and eggs) and can be transmitted to subsequent generations, somatic cell genetic disorders arise only in specific somatic cells.Dominant(dominancy): the trait expressed in heterozygotes (Aa).Recessive(recessivity): the trait expressed only in homozygotes (aa).。
.WorkshopAlgebraic Methods inFunctional AnalysisList of Participants,Schedule and Abstracts of TalksMathematical Sciences,Chalmers University of Technology andG¨o teborg University.WorkshopAlgebraic Methods inFunctional AnalysisMathematical Sciences,Chalmers University of Technology andG¨o teborg UniversityGothenburg,SWEDENJune15–17,2007Supported by The Swedish Foundation for International Cooperation in Research and Higher Education(STINT)1Organizers:•Volodymyr Mazorchuk,Department of Mathematics,Uppsala Uni-versity;•Lyudmila Turowska,Mathematical Sciences,Chalmers University of Technology and G¨o teborg University2List of Participants:•Eshaghi Gorji Majid,Semnan University,Semnan,IRAN;•Iusenko Kostyantyn,Institute of Mathematics,Kyiv,UKRAINE;•Juschenko Kate,Chalmers University Technology and G¨o teborg Uni-versity,SWEDEN;•Lattarulo Michele,Universit´a di Genova,ITALY;•Levene Rupert,Queen’s University,Belfast,UK;•Ludwig Jean,Metz University,FRANCE;•Mathieu Martin,Queen’s University,Belfast,UK;•Mazorchuk Volodymyr,University of Uppsala,SWEDEN;•Nest Ryszard,Copenhagen University,DENMARK;•Neshveyev Sergey,Oslo University,NORWAY•Ortega Eduard,Universitat Aut´o noma de Barcelona,SPAIN;•Irina Peterburgsky,Suffolk University,Boston,US•Popovych Stanislav,Chalmers University Technology and G¨o teborg University,SWEDEN;•Proskurin Daniel,Kyiv Taras Shevchenko University,UKRAINE;•Rørdam Mikael,University of Southern Denmark,Odense,DEN-MARK;•Savchuk Yurii,Max Planck Institute f¨u r Mathematik in den Natur-wissenschaften;GERMANY;•Sj¨o gren Peter,Chalmers University Technology and Gteborg Univer-sity,SWEDEN;•Todorov Ivan Queen’s University,Belfast,UK;•Turowska Lyudmila,Chalmers University Technology and Gteborg University,SWEDEN;3•Vinh Le Anh Harvard University,US•Zhang Genkai,Chalmers University Technology and Gteborg Uni-versity,SWEDEN;4Schedule of TalksSaturday,June16-th,2007.10.00-10.50Sergey Neshveev Dirac operators on compact quantum groups11.00-11.30Coffee/Tea Break11.30-11.55Stanislav Popovych Matrix Ordered Operator Algebras12.00-14.00Lunch14.00-14.50Mikael Rørdam On the structure of C(X)-algebras15.00-15.25Ivan Todorov Operator ranges and C∗-algebras15.30-16.00Coffee/Tea Break16.00-16.30Yurii Savchuk Non-commutative analogues of17th Hilbert problem.16.30-17.30Ryszard Nest Existence and classification of deformations of gerbes.19.00Conference Dinner Vivaldi Restaurang,Berzeliigatan19Sunday,June17-th,2007.09.30-10.20Martin Mathieu The structure of Lie derivations10.30-10.55Majid Eshaghi Gorji N-ideal amenability of Banach algebras11.00-11.30Coffee/Tea Break11.30-11.55Kate Juschenko/Operator multipliersLyudmila Turowska12.00-12.50Jean Ludwig Simple modules of some group algebras5AbstractsN-ideal amenability of Banach algebrasMajid Eshaghi GorjiSemnan University,IranWe introduce two notions of amenability for a Banach algebra A.Let n∈N and let I be a closed two-sided ideal in A,A is n−I−weakly amenable if thefirst cohomology group of A with coefficients in the n-th dual space I(n)is zero;i.e.,H1(A,I(n))={0}.Further,A is n-ideally amenable if A is n−I−weakly amenable for every closed two-sided ideal I in A.We study the n-ideal amenability of some classes of Banach algebras.We show that B(H)is n-ideally amenable for every n∈N and for every Hilbert space H. We show that every C∗−algebra is n-ideally amenable for n=2k+1.We study the n-ideal amenability of commutative Banach algebras.”Simple modules of some group algebrasJean LudwigMetz University,FranceWe determine the simple modules of the L1-algebras of Heisenberg’s and Boidol’s group and of Sl(2,R).We show that up to equivalence these modules are thefinite rank submodules of the L p principal series(and of the discrete series representations in the case of SL2(R)).The same result holds for every exponential Lie group.We derive necessary and sufficient conditions for an ambiskew polynomial ring to have a Hopf algebra structure of a certain type.This construction generalizes many known Hopf algebras,for example U(sl2),U q(sl2)and the enveloping algebra of the3-dimensional Heisenberg Lie algebra.In a torsion-free case we describe thefinite-dimensional simple modules,in particular their dimensions and prove a Clebsch-Gordan decomposition theorem for the tensor product of two simple modules.We construct a Casimir type operator and prove that anyfinite-dimensional weight module is semisimple.6The structure of Lie derivationsMartin MathieuQueen’s University,Belfast,UKThe structure of Lie derivations on a C∗-algebra has a surprisingly simple pattern.Yet,to establish this result,which was completed in joint work with Armando Villena(Granada),a substantial amount of theory had to be developed.In fact,this was one of the main motivations to investigate local multipliers of C∗-algebras in our monograph with Pere Ara(Barcelona).The approach is essentially algebraic with a few analytic tweaks;these,however, turned out to be fairly tricky at times.In our talk we plan to discuss this work in some detail with the overall theme of the workshop in mind. Dirac operators on compact quantum groupsSergey NeshveevOslo University,NorwayFor the q-deformation G q,0<q<1,of any simply connected simple compact Lie group G we construct an equivariant spectral triple which is an isospectral deformation of that defined by the Dirac operator D on G.The construction depends on the choice of a twist,which can be thought of as a2-cochain on the dual discrete quantum groupˆG.It turns out,the key properties of our quantum Dirac operators depend not on the twist but on the associator,that is,the corresponding coboundary onˆG.What allows us to say something nontrivial about the quantum Dirac operators,is that by results of Drinfeld and Kazhdan-Lusztig we can alwaysfind a twist such that the corresponding associator is determined by the monodromy of a system of partial differential equations.(Joint work with Lars Tuset.)7Existence and classification of deformationsof gerbes.Ryszard NestCopenhagen University,DenmarkIn this talk we will study deformation quantization of gerbes.After ba-sic definitions we will interpret deformations of a stack as Maurer-Cartan elements of a differential graded Lie algebra and classify deformations of a given gerbe in terms of Maurer-Cartan elements of the DGLA of Hochschild cochains twisted by the cohomology class of the gerbe.In particular we will get a classification of all deformations of a given gerbe on a symplectic manifold.Operators on Spaces of Abstract Valued Functions and Their Norms.SomeApplications.Irina PeterburgskySuffolk University,Boston,USWe proved that under certain conditions norms of linear operators over corresponding classes of scalar valued and Hilbert or Banach space valued functions coincide.Various applications of this general fact were found.In particular,extremal problems for norms of linear operators over spaces of analytic functions in several variables were ing our technique, we generalized ndau coefficient problem for a case of Hilbert or Banach codomain space.8Matrix Ordered Operator AlgebrasEkaterina Juschenko,Stanislav Popovych Chalmers University of Technology and G¨o teborg University We present a characterization of C∗-representability of an arbitrary∗-algebra in terms of algebraically admissible cones.It is analogues to Choi and Effros characterization of abstract operator systems.Then we discuss a question when for a given∗-algebra A a sequence of cones C n∈M n(A)can be realized as cones of positive operators in a faithful∗-representation of A on a Hilbert space.As an application of the above results we present a char-acterization of operator algebras which are completely boundedly isomorphic to C∗-algebras.Some connections with Kadison’s Similarity problem will be discussed.References[1]M.D.Choi,E.G.Effros,Injectivity and operator spaces.J.FunctionalAnalysis24(1977),no.2,156–209.[2]E.Juschenko,S.Popovych,Matrix Ordered Operator Algebras.,Chalmers&G¨o teborg University math.preprint2007:9.[3]S.Popovych,On O∗-representability and C∗-representability of∗-algebras.Chalmers&G¨o teborg University math.preprint2006:35.On the structure of C(X)-algebrasMikael RørdamUniversity of Southern Denmark,OdenseC(X)-algebras form a special class of non-simple C-algebras that extends the class of continuousfield C-algebras.A C(X)-algebra is“assembled”over a compact Hausdorffspace X fromfibre algebras(one for each point in the space X).First,we shall spend some time giving the appropriate definitions and looking at examples.Then we shall address a number of recent results that describe when given properties(eg.stability,tensorially absorbing certain C-algebras,being properly infinite)of thefibre algebra9pass to the C(X)-algebra itself.In many cases one has nice results when the space X hasfinite dimension,and“counterexamples”when the space X has infinite dimension.Non-commutative analogues of17th HilbertproblemYurii Savchuk,Max Planck Institute f¨u r Mathematik in denNaturwissenschaften,GERMANYA self-adjoint element c of a∗-algebra A is called positive ifπ(c)is positive operator for all”good”∗-representations of A.For some∗-algebras we prove the Positivstellensatz:for every positive element c there exist elements x=0,x1,...,x n such that x∗cx=x∗1x1+ (x)nx n.The Positivstellensatz foralgebra of polynomials C[t1,...,t k]is the Artin’s solution to17th.Hilbert problem.Operator ranges and C*-algebrasIvan G.TodorovQueen’s University,Belfast,UKThe collection of all ranges of operators on a Hilbert space or,more gen-erally,in a given von Neumann algebra,is a lattice with respect to intersec-tion and(non-closed)linear span.This property does not hold for arbitrary C*-algebras of operators.Moreover,given a C*-algebra A and a faithful representationπof A,whether or notπ(A)possesses this property depends onπ.Say thatπpossesses property L if the operator ranges fromπ(A)form a lattice under the above operations.The talk will be concerned with the study of the above property.In particular,property L will be related to the so called“directed set property”of a C*-algebra A,namely the property that the set of allfinite dimensional C*-subalgebras of A be directed by inclusion.A theorem on the structure of the collection of all representations of A possessing L will be discussed.The talk will be based on a joint work with M.Anoussis(Samos)and A. Katavolos(Athens)10Operator multipliersKate Juschenko/Lyudmila TurowskaChalmers University of Technology and G¨o teborg University,SwedenOperator multipliers were recently introduced by Kissin ad Shulman as a non-commutative version of Schur multipliers.They are elements of the minimal tensor product of two C∗-algebras satisfying certain boundedness conditions.In this talk we will descuss certain universal operator multipliers.We es-tablish a non-commutative version of the characterisations by Grothendieck and Peller which shows that universal operator multipliers can be obtained as certain weak limit of elements of the algebraic tensor product of the cor-responding C∗-algebras.The talk will be based on a joint work with Ivan Todorov(Belfast).11。
《材料科学与工程基础》英文习题及思考题及答案第二章习题和思考题Questions and Problems2.6 Allowed values for the quantum numbers ofelectrons are as follows:The relationships between n and the shell designationsare noted in Table 2.1.Relative tothe subshells,l ＝0 corresponds to an s subshelll ＝1 corresponds to a p subshelll ＝2 corresponds to a d subshelll ＝3 corresponds to an f subshellFor the K shell, the four quantum numbersfor each of the two electrons in the 1s state, inthe order of nlmlms , are 100(1/2 ) and 100(-1/2 ).Write the four quantum numbers for allof the electrons inthe L and M shells, and notewhich correspond to the s, p, and d subshells.2.7 Give the electron configurations for the followingions: Fe2+, Fe3+, Cu+, Ba2+,Br-, andS2-.2.17 (a) Briefly cite the main differences betweenionic, covalent, and metallicbonding.(b) State the Pauli exclusion principle.2.18 Offer an explanation as to why covalently bonded materials are generally lessdense than ionically or metallically bonded ones.2.19 Compute the percents ionic character of the interatomic bonds for the followingcompounds: TiO2 , ZnTe, CsCl, InSb, and MgCl2 .2.21 Using Table 2.2, determine the number of covalent bonds that are possible foratoms of the following elements: germanium, phosphorus, selenium, and chlorine.2.24 On the basis of the hydrogen bond, explain the anomalous behavior of waterwhen it freezes. That is, why is there volume expansion upon solidification?3.1 What is the difference between atomic structure and crystal structure?3.2 What is the difference between a crystal structure and a crystal system?3.4Show for the body-centered cubic crystal structure that the unit cell edge lengtha and the atomic radius R are related through a =4R/√3.3.6 Show that the atomic packing factor for BCC is 0.68. .3.27* Show that the minimum cation-to-anion radius ratio for a coordinationnumber of 6 is 0.414. Hint: Use the NaCl crystal structure (Figure 3.5), and assume that anions and cations are just touching along cube edges and across face diagonals.3.48 Draw an orthorhombic unit cell, and within that cell a [121] direction and a(210) plane.3.50 Here are unit cells for two hypothetical metals:(a)What are the indices for the directions indicated by the two vectors in sketch(a)?(b) What are the indices for the two planes drawn in sketch (b)?3.51* Within a cubic unit cell, sketch the following directions:.3.53 Determine the indices for the directions shown in the following cubic unit cell:3.57 Determine the Miller indices for the planesshown in the following unit cell:3.58Determine the Miller indices for the planes shown in the following unit cell: 3.61* Sketch within a cubic unit cell the following planes:3.62 Sketch the atomic packing of (a) the (100)plane for the FCC crystal structure, and (b) the (111) plane for the BCC crystal structure (similar to Figures 3.24b and 3.25b).3.77 Explain why the properties of polycrystalline materials are most oftenisotropic.5.1 Calculate the fraction of atom sites that are vacant for lead at its meltingtemperature of 327_C. Assume an energy for vacancy formation of 0.55eV/atom.5.7 If cupric oxide (CuO) is exposed to reducing atmospheres at elevatedtemperatures, some of the Cu2_ ions will become Cu_.(a) Under these conditions, name one crystalline defect that you would expect toform in order to maintain charge neutrality.(b) How many Cu_ ions are required for the creation of each defect?5.8 Below, atomic radius, crystal structure, electronegativity, and the most commonvalence are tabulated, for several elements; for those that are nonmetals, only atomic radii are indicated.Which of these elements would you expect to form the following with copper:(a) A substitutional solid solution having complete solubility?(b) A substitutional solid solution of incomplete solubility?(c) An interstitial solid solution?5.9 For both FCC and BCC crystal structures, there are two different types ofinterstitial sites. In each case, one site is larger than the other, which site isnormally occupied by impurity atoms. For FCC, this larger one is located at the center of each edge of the unit cell; it is termed an octahedral interstitial site. On the other hand, with BCC the larger site type is found at 0, __, __ positions—that is, lying on _100_ faces, and situated midway between two unit cell edges on this face and one-quarter of the distance between the other two unit cell edges; it is termed a tetrahedral interstitial site. For both FCC and BCC crystalstructures, compute the radius r of an impurity atom that will just fit into one of these sites in terms of the atomic radius R of the host atom.5.10 (a) Suppose that Li2O is added as an impurity to CaO. If the Li_ substitutes forCa2_, what kind of vacancies would you expect to form? How many of thesevacancies are created for every Li_ added?(b) Suppose that CaCl2 is added as an impurity to CaO. If the Cl_ substitutes forO2_, what kind of vacancies would you expect to form? How many of thevacancies are created for every Cl_ added?5.28 Copper and platinum both have the FCC crystal structure and Cu forms asubstitutional solid solution for concentrations up to approximately 6 wt% Cu at room temperature. Compute the unit cell edge length for a 95 wt% Pt-5 wt% Cu alloy.5.29 Cite the relative Burgers vector–dislocation line orientations for edge, screw, andmixed dislocations.6.1 Briefly explain the difference between selfdiffusion and interdiffusion.6.3 (a) Compare interstitial and vacancy atomic mechanisms for diffusion.(b) Cite two reasons why interstitial diffusion is normally more rapid thanvacancy diffusion.6.4 Briefly explain the concept of steady state as it applies to diffusion.6.5 (a) Briefly explain the concept of a driving force.(b) What is the driving force for steadystate diffusion?6.6Compute the number of kilograms of hydrogen that pass per hour through a5-mm thick sheet of palladium having an area of 0.20 m2at 500℃. Assume a diffusion coefficient of 1.0×10- 8 m2/s, that the concentrations at the high- and low-pressure sides of the plate are 2.4 and 0.6 kg of hydrogen per cubic meter of palladium, and that steady-state conditions have been attained.6.7 A sheet of steel 1.5 mm thick has nitrogen atmospheres on both sides at 1200℃and is permitted to achieve a steady-state diffusion condition. The diffusion coefficient for nitrogen in steel at this temperature is 6×10-11m2/s, and the diffusion flux is found to be 1.2×10- 7kg/m2-s. Also, it is known that the concentration of nitrogen in the steel at the high-pressure surface is 4 kg/m3. How far into the sheet from this high-pressure side will the concentration be 2.0 kg/m3?Assume a linear concentration profile.6.24. Carbon is allowed to diffuse through a steel plate 15 mm thick. Theconcentrations of carbon at the two faces are 0.65 and 0.30 kg C/m3 Fe, whichare maintained constant. If the preexponential and activation energy are 6.2 _10_7 m2/s and 80,000 J/mol, respectively, compute the temperature at which the diffusion flux is 1.43 _ 10_9 kg/m2-s.6.25 The steady-state diffusion flux through a metal plate is 5.4_10_10 kg/m2-s at atemperature of 727_C (1000 K) and when the concentration gradient is _350kg/m4. Calculate the diffusion flux at 1027_C (1300 K) for the sameconcentration gradient and assuming an activation energy for diffusion of125,000 J/mol.10.2 What thermodynamic condition must be met for a state of equilibrium to exist? 10.4 What is the difference between the states of phase equilibrium and metastability?10.5 Cite the phases that are present and the phase compositions for the followingalloys:(a) 90 wt% Zn–10 wt% Cu at 400℃(b) 75 wt% Sn–25wt%Pb at 175℃(c) 55 wt% Ag–45 wt% Cu at 900℃(d) 30 wt% Pb–70 wt% Mg at 425℃(e) 2.12 kg Zn and 1.88 kg Cu at 500℃(f ) 37 lbm Pb and 6.5 lbm Mg at 400℃(g) 8.2 mol Ni and 4.3 mol Cu at 1250℃.(h) 4.5 mol Sn and 0.45 mol Pb at 200℃10.6 For an alloy of composition 74 wt% Zn–26 wt% Cu, cite the phases presentand their compositions at the following temperatures: 850℃, 750℃, 680℃, 600℃, and 500℃.10.7 Determine the relative amounts (in terms of mass fractions) of the phases forthe alloys and temperatures given inProblem 10.5.10.9 Determine the relative amounts (interms of volume fractions) of the phases forthe alloys and temperatures given inProblem 10.5a, b, and c. Below are given theapproximate densities of the various metalsat the alloy temperatures:10.18 Is it possible to have a copper–silveralloy that, at equilibrium, consists of a _ phase of composition 92 wt% Ag–8wt% Cu, and also a liquid phase of composition 76 wt% Ag–24 wt% Cu? If so, what will be the approximate temperature of the alloy? If this is not possible,explain why.10.20 A copper–nickel alloy of composition 70 wt% Ni–30 wt% Cu is slowly heatedfrom a temperature of 1300_C .(a) At what temperature does the first liquid phase form?(b) What is the composition of this liquid phase?(c) At what temperature does complete melting of the alloy occur?(d) What is the composition of the last solid remaining prior to completemelting?10.28 .Is it possible to have a copper–silver alloy of composition 50 wt% Ag–50 wt%Cu, which, at equilibrium, consists of _ and _ phases having mass fractions W_ _0.60 and W_ _ 0.40? If so, what will be the approximate temperature of the alloy?If such an alloy is not possible, explain why.10.30 At 700_C , what is the maximum solubility (a) of Cu in Ag? (b) Of Ag in Cu?第三章习题和思考题3.3If the atomic radius of aluminum is 0.143nm, calculate the volume of its unitcell in cubic meters.3.8 Iron has a BCC crystal structure, an atomic radius of 0.124 nm, and an atomicweight of 55.85 g/mol. Compute and compare its density with the experimental value found inside the front cover.3.9 Calculate the radius of an iridium atom given that Ir has an FCC crystal structure,a density of 22.4 g/cm3, and an atomic weight of 192.2 g/mol.3.13 Using atomic weight, crystal structure, and atomic radius data tabulated insidethe front cover, compute the theoretical densities of lead, chromium, copper, and cobalt, and then compare these values with the measured densities listed in this same table. The c/a ratio for cobalt is 1.623.3.15 Below are listed the atomic weight, density, and atomic radius for threehypothetical alloys. For each determine whether its crystal structure is FCC,BCC, or simple cubic and then justify your determination. A simple cubic unitcell is shown in Figure 3.40.3.21 This is a unit cell for a hypotheticalmetal:(a) To which crystal system doesthis unit cell belong?(b) What would this crystal structure be called?(c) Calculate the density of the material, given that its atomic weight is 141g/mol.3.25 For a ceramic compound, what are the two characteristics of the component ionsthat determine the crystal structure?3.29 On the basis of ionic charge and ionic radii, predict the crystal structures for thefollowing materials: (a) CsI, (b) NiO, (c) KI, and (d) NiS. Justify your selections.3.35 Magnesium oxide has the rock salt crystal structure and a density of 3.58 g/cm3.(a) Determine the unit cell edge length. (b) How does this result compare withthe edge length as determined from the radii in Table 3.4, assuming that theMg2_ and O2_ ions just touch each other along the edges?3.36 Compute the theoretical density of diamond given that the CUC distance andbond angle are 0.154 nm and 109.5°, respectively. How does this value compare with the measured density?3.38 Cadmium sulfide (CdS) has a cubic unit cell, and from x-ray diffraction data it isknown that the cell edge length is 0.582 nm. If the measured density is 4.82 g/cm3 , how many Cd 2+ and S 2—ions are there per unit cell?3.41 A hypothetical AX type of ceramic material is known to have a density of 2.65g/cm 3 and a unit cell of cubic symmetry with a cell edge length of 0.43 nm. The atomic weights of the A and X elements are 86.6 and 40.3 g/mol, respectively.On the basis of this information, which of the following crystal structures is (are) possible for this material: rock salt, cesium chloride, or zinc blende? Justify your choice(s).3.42 The unit cell for Mg Fe2O3 (MgO-Fe2O3) has cubic symmetry with a unit celledge length of 0.836 nm. If the density of this material is 4.52 g/cm 3 , compute its atomic packing factor. For this computation, you will need to use ionic radii listed in Table 3.4.3.44 Compute the atomic packing factor for the diamond cubic crystal structure(Figure 3.16). Assume that bonding atoms touch one another, that the angle between adjacent bonds is 109.5°, and that each atom internal to the unit cell is positioned a/4 of the distance away from the two nearest cell faces (a is the unit cell edge length).3.45 Compute the atomic packing factor for cesium chloride using the ionic radii inTable 3.4 and assuming that the ions touch along the cube diagonals.3.46 In terms of bonding, explain why silicate materials have relatively low densities.3.47 Determine the angle between covalent bonds in an SiO44—tetrahedron.3.63 For each of the following crystal structures, represent the indicated plane in themanner of Figures 3.24 and 3.25, showing both anions and cations: (a) (100)plane for the rock salt crystal structure, (b) (110) plane for the cesium chloride crystal structure, (c) (111) plane for the zinc blende crystal structure, and (d) (110) plane for the perovskite crystal structure.3.66 The zinc blende crystal structure is one that may be generated from close-packedplanes of anions.(a) Will the stacking sequence for this structure be FCC or HCP? Why?(b) Will cations fill tetrahedral or octahedral positions? Why?(c) What fraction of the positions will be occupied?3.81* The metal iridium has an FCC crystal structure. If the angle of diffraction forthe (220) set of planes occurs at 69.22°(first-order reflection) when monochromatic x-radiation having a wavelength of 0.1542 nm is used, compute(a) the interplanar spacing for this set of planes, and (b) the atomic radius for aniridium atom.4.10 What is the difference between configuration and conformation in relation topolymer chains? vinyl chloride).4.22 (a) Determine the ratio of butadiene to styrene mers in a copolymer having aweight-average molecular weight of 350,000 g/mol and weight-average degree of polymerization of 4425.(b) Which type(s) of copolymer(s) will this copolymer be, considering thefollowing possibilities: random, alternating, graft, and block? Why?4.23 Crosslinked copolymers consisting of 60 wt% ethylene and 40 wt% propylenemay have elastic properties similar to those for natural rubber. For a copolymer of this composition, determine the fraction of both mer types.4.25 (a) Compare the crystalline state in metals and polymers.(b) Compare thenoncrystalline state as it applies to polymers and ceramic glasses.4.26 Explain briefly why the tendency of a polymer to crystallize decreases withincreasing molecular weight.4.27* For each of the following pairs of polymers, do the following: (1) state whetheror not it is possible to determine if one polymer is more likely to crystallize than the other; (2) if it is possible, note which is the more likely and then cite reason(s) for your choice; and (3) if it is not possible to decide, then state why.(a) Linear and syndiotactic polyvinyl chloride; linear and isotactic polystyrene.(b) Network phenol-formaldehyde; linear and heavily crosslinked ci s-isoprene.(c) Linear polyethylene; lightly branched isotactic polypropylene.(d) Alternating poly(styrene-ethylene) copolymer; randompoly(vinylchloride-tetrafluoroethylene) copolymer.4.28 Compute the density of totally crystalline polyethylene. The orthorhombic unitcell for polyethylene is shown in Figure 4.10; also, the equivalent of two ethylene mer units is contained within each unit cell.5.11 What point defects are possible for MgO as an impurity in Al2O3? How manyMg 2+ ions must be added to form each of these defects?5.13 What is the composition, in weight percent, of an alloy that consists of 6 at% Pband 94 at% Sn?5.14 Calculate the composition, in weight per-cent, of an alloy that contains 218.0 kgtitanium, 14.6 kg of aluminum, and 9.7 kg of vanadium.5.23 Gold forms a substitutional solid solution with silver. Compute the number ofgold atoms per cubic centimeter for a silver-gold alloy that contains 10 wt% Au and 90 wt% Ag. The densities of pure gold and silver are 19.32 and 10.49 g/cm3 , respectively.8.53 In terms of molecular structure, explain why phenol-formaldehyde (Bakelite)will not be an elastomer.10.50 Compute the mass fractions of αferrite and cementite in pearlite. assumingthat pressure is held constant.10.52 (a) What is the distinction between hypoeutectoid and hypereutectoid steels?(b) In a hypoeutectoid steel, both eutectoid and proeutectoid ferrite exist. Explainthe difference between them. What will be the carbon concentration in each?10.56 Consider 1.0 kg of austenite containing 1.15 wt% C, cooled to below 727_C(a) What is the proeutectoid phase?(b) How many kilograms each of total ferrite and cementite form?(c) How many kilograms each of pearlite and the proeutectoid phase form?(d) Schematically sketch and label the resulting microstructure.10.60 The mass fractions of total ferrite and total cementite in an iron–carbon alloyare 0.88 and 0.12, respectively. Is this a hypoeutectoid or hypereutectoid alloy?Why?10.64 Is it possible to have an iron–carbon alloy for which the mass fractions of totalferrite and proeutectoid cementite are 0.846 and 0.049, respectively? Why orwhy not?第四章习题和思考题7.3 A specimen of aluminum having a rectangular cross section 10 mm _ 12.7 mmis pulled in tension with 35,500 N force, producing only elastic deformation. 7.5 A steel bar 100 mm long and having a square cross section 20 mm on an edge ispulled in tension with a load of 89,000 N , and experiences an elongation of 0.10 mm . Assuming that the deformation is entirely elastic, calculate the elasticmodulus of the steel.7.7 For a bronze alloy, the stress at which plastic deformation begins is 275 MPa ,and the modulus of elasticity is 115 Gpa .(a) What is the maximum load that may be applied to a specimen with across-sectional area of 325mm, without plastic deformation?(b) If the original specimen length is 115 mm , what is the maximum length towhich it may be stretched without causing plastic deformation?7.8 A cylindrical rod of copper (E _ 110 GPa, Stress (MPa) ) having a yield strengthof 240Mpa is to be subjected to a load of 6660 N. If the length of the rod is 380 mm, what must be the diameter to allow an elongation of 0.50 mm?7.9 Consider a cylindrical specimen of a steel alloy (Figure 7.33) 10mm in diameterand 75 mm long that is pulled in tension. Determine its elongation when a load of 23,500 N is applied.7.16 A cylindrical specimen of some alloy 8 mm in diameter is stressed elasticallyin tension. A force of 15,700 N produces a reduction in specimen diameter of 5 _ 10_3 mm. Compute Poisson’s ratio for this material if its modulus of elasticity is 140 GPa .7.17 A cylindrical specimen of a hypothetical metal alloy is stressed in compression.If its original and final diameters are 20.000 and 20.025 mm, respectively, and its final length is 74.96 mm, compute its original length if the deformation is totally elastic. The elastic and shear moduli for this alloy are 105 Gpa and 39.7 GPa,respectively.7.19 A brass alloy is known to have a yield strength of 275 MPa, a tensile strength of380 MPa, and an elastic modulus of 103 GPa . A cylindrical specimen of thisalloy 12.7 mm in diameter and 250 mm long is stressed in tension and found to elongate 7.6 mm . On the basis of the information given, is it possible tocompute the magnitude of the load that is necessary to produce this change inlength? If so, calculate the load. If not, explain why.7.20A cylindrical metal specimen 15.0mmin diameter and 150mm long is to besubjected to a tensile stress of 50 Mpa; at this stress level the resulting deformation will be totally elastic.(a)If the elongation must be less than 0.072mm,which of the metals in Tabla7.1are suitable candidates? Why ?(b)If, in addition, the maximum permissible diameter decrease is 2.3×10-3mm,which of the metals in Table 7.1may be used ? Why?7.22 Cite the primary differences between elastic, anelastic, and plastic deformationbehaviors.7.23 diameter of 10.0 mm is to be deformed using a tensile load of 27,500 N. It mustnot experience either plastic deformation or a diameter reduction of more than7.5×10-3 mm. Of the materials listed as follows, which are possible candidates?Justify your choice(s).7.24 A cylindrical rod 380 mm long, having a diameter of 10.0 mm, is to besubjected to a tensile load. If the rod is to experience neither plastic deformationnor an elongation of more than 0.9 mm when the applied load is 24,500 N,which of the four metals or alloys listed below are possible candidates?7.25 Figure 7.33 shows the tensile engineering stress–strain behavior for a steel alloy.(a) What is the modulus of elasticity?(b) What is the proportional limit?(c) What is the yield strength at a strain offset of 0.002?(d) What is the tensile strength?7.27 A load of 44,500 N is applied to a cylindrical specimen of steel (displaying thestress–strain behavior shown in Figure 7.33) that has a cross-sectional diameter of 10 mm .(a) Will the specimen experience elastic or plastic deformation? Why?(b) If the original specimen length is 500 mm), how much will it increase inlength when t his load is applied?7.29 A cylindrical specimen of aluminumhaving a diameter of 12.8 mm and a gaugelength of 50.800 mm is pulled in tension. Usethe load–elongation characteristics tabulatedbelow to complete problems a through f.(a)Plot the data as engineering stressversusengineering strain.(b) Compute the modulus of elasticity.(c) Determine the yield strength at astrainoffset of 0.002.(d) Determine the tensile strength of thisalloy.(e) What is the approximate ductility, in percent elongation?(f ) Compute the modulus of resilience.7.35 (a) Make a schematic plot showing the tensile true stress–strain behavior for atypical metal alloy.(b) Superimpose on this plot a schematic curve for the compressive truestress–strain behavior for the same alloy. Explain any difference between thiscurve and the one in part a.(c) Now superimpose a schematic curve for the compressive engineeringstress–strain behavior for this same alloy, and explain any difference between this curve and the one in part b.7.39 A tensile test is performed on a metal specimen, and it is found that a true plasticstrain of 0.20 is produced when a true stress of 575 MPa is applied; for the same metal, the value of K in Equation 7.19 is 860 MPa. Calculate the true strain that results from the application of a true stress of 600 Mpa.7.40 For some metal alloy, a true stress of 415 MPa produces a plastic true strain of0.475. How much will a specimen of this material elongate when a true stress of325 MPa is applied if the original length is 300 mm ? Assume a value of 0.25 for the strain-hardening exponent n.7.43 Find the toughness (or energy to cause fracture) for a metal that experiences bothelastic and plastic deformation. Assume Equation 7.5 for elastic deformation,that the modulus of elasticity is 172 GPa , and that elastic deformation terminates at a strain of 0.01. For plastic deformation, assume that the relationship between stress and strain is described by Equation 7.19, in which the values for K and n are 6900 Mpa and 0.30, respectively. Furthermore, plastic deformation occurs between strain values of 0.01 and 0.75, at which point fracture occurs.7.47 A steel specimen having a rectangular cross section of dimensions 19 mm×3.2mm (0.75in×0.125in.) has the stress–strain behavior shown in Figure 7.33. If this specimen is subjected to a tensile force of 33,400 N (7,500lbf ), then(a) Determine the elastic and plastic strain values.(b) If its original length is 460 mm (18 in.), what will be its final length after theload in part a is applied and then released?7.50 A three-point bending test was performed on an aluminum oxide specimenhaving a circular cross section of radius 3.5 mm; the specimen fractured at a load of 950 N when the distance between the support points was 50 mm . Another test is to be performed on a specimen of this same material, but one that has a square cross section of 12 mm length on each edge. At what load would you expect this specimen to fracture if the support point separation is 40 mm ?7.51 (a) A three-point transverse bending test is conducted on a cylindrical specimenof aluminum oxide having a reported flexural strength of 390 MPa . If the speci- men radius is 2.5 mm and the support point separation distance is 30 mm ,predict whether or not you would expect the specimen to fracture when a load of 620 N is applied. Justify your prediction.(b) Would you be 100% certain of the prediction in part a? Why or why not?7.57 When citing the ductility as percent elongation for semicrystalline polymers, it isnot necessary to specify the specimen gauge length, as is the case with metals.Why is this so?7.66 Using the data represented in Figure 7.31, specify equations relating tensilestrength and Brinell hardness for brass and nodular cast iron, similar toEquations 7.25a and 7.25b for steels.8.4 For each of edge, screw, and mixed dislocations, cite the relationship between thedirection of the applied shear stress and the direction of dislocation line motion.8.5 (a) Define a slip system.(b) Do all metals have the same slip system? Why or why not?8.7. One slip system for theBCCcrystal structure is _110__111_. In a manner similarto Figure 8.6b sketch a _110_-type plane for the BCC structure, representingatom positions with circles. Now, using arrows, indicate two different _111_ slip directions within this plane.8.15* List four major differences between deformation by twinning and deformationby slip relative to mechanism, conditions of occurrence, and final result.8.18 Describe in your own words the three strengthening mechanisms discussed inthis chapter (i.e., grain size reduction, solid solution strengthening, and strainhardening). Be sure to explain how dislocations are involved in each of thestrengthening techniques.8.19 (a) From the plot of yield strength versus (grain diameter)_1/2 for a 70 Cu–30 Zncartridge brass, Figure 8.15, determine values for the constants _0 and ky inEquation 8.5.(b) Now predict the yield strength of this alloy when the average grain diameteris 1.0 _ 10_3 mm.8.20. The lower yield point for an iron that has an average grain diameter of 5 _ 10_2mm is 135 MPa . At a grain diameter of 8 _ 10_3 mm, the yield point increases to 260MPa. At what grain diameter will the lower yield point be 205 Mpa ?8.24 (a) Show, for a tensile test, thatif there is no change in specimen volume during the deformation process (i.e., A0 l0 _Ad ld).(b) Using the result of part a, compute the percent cold work experienced bynaval brass (the stress–strain behavior of which is shown in Figure 7.12) when a stress of 400 MPa is applied.8.25 Two previously undeformed cylindrical specimens of an alloy are to be strainhardened by reducing their cross-sectional areas (while maintaining their circular cross sections). For one specimen, the initial and deformed radii are 16 mm and11 mm, respectively. The second specimen, with an initial radius of 12 mm, musthave the same deformed hardness as the first specimen; compute the secondspecimen’s radius after deformation.8.26 Two previously undeformed specimens of the same metal are to be plasticallydeformed by reducing their cross-sectional areas. One has a circular cross section, and the other is rectangular is to remain as such. Their original and deformeddimensions are as follows:Which of these specimens will be the hardest after plastic deformation, and why?8.27 A cylindrical specimen of cold-worked copper has a ductility (%EL) of 25%. Ifits coldworked radius is 10 mm (0.40 in.), what was its radius beforedeformation?8.28 (a) What is the approximate ductility (%EL) of a brass that has a yield strengthof 275 MPa ?(b) What is the approximate Brinell hardness of a 1040 steel having a yieldstrength of 690 MPa?8.41 In your own words, describe the mechanisms by which semicrystalline polymers(a) elasticallydeform and (b) plastically deform, and (c) by which elastomerselastically deform.8.42 Briefly explain how each of the following influences the tensile modulus of asemicrystallinepolymer and why:(a) molecular weight;(b) degree of crystallinity;(c) deformation by drawing;(d) annealing of an undeformed material;(e) annealing of a drawn material.8.43* Briefly explain how each of the following influences the tensile or yieldstrength of a semicrystalline polymer and why:(a) molecular weight;。
a rX iv:mat h /98234v1[mat h.FA]6Feb1998DEFORMATION QUANTIZATION OF CERTAIN NON-LINEAR POISSON STRUCTURES BYUNG–JAY KAHNG Abstract.As a generalization of the linear Poisson bracket on the dual space of a Lie algebra,we introduce certain non-linear Poisson brackets which are “cocycle perturbations”of the linear Poisson bracket.We show that these special Poisson brackets are equivalent to Poisson brackets of central extension type,which re-semble the central extensions of an ordinary Lie bracket via Lie algebra cocycles.We are able to formulate (strict)deformation quantizations of these Poisson brackets by means of twisted group C ∗–algebras.We also indicate that these deformation quantiza-tions can be used to construct some specific non-compact quantum groups.Introduction.Let M be a Poisson manifold.Consider C ∞(M ),the commutative algebra under pointwise multiplication of smooth functions on M .We attempt to deform the pointwise product of smooth functions into a noncommutative product,with respect to a parameter ,such that the direction of the deformation is given by the Poisson bracket on M .This problem of finding a deformation quanti-zation of M ([33],[1])is actually a problem dating back to the early days of quantum mechanics [34],[20].We are particularly interested in the settings where the deformed product of functions is again a function—in contrast to much of the literature on the subject involving formal power series,or the so-called “star products”.In this direction,Rieffel has been developing the no-tion of “strict”deformation quantization,in the C ∗–algebra framework[26,29].Here,in addition to the requirement that the deformed prod-uct of functions is again a function,the deformed algebra is required to have an involution and a C ∗–norm.By using the C ∗–algebra frame-work,one gains the advantage of being able to keep the topological and geometric aspects of the given manifold while we perform the quanti-zation.Let h be a (finite dimensional)Lie algebra.It is well-known [35]that the Lie algebra structure on h defines a natural Poisson bracket on the dual vector space h ∗of h ,which is called a linear Poisson bracket .2BYUNG–JAY KAHNGThis Poisson bracket is also called the“Lie–Poisson bracket”,to em-phasize the fact that it actually was already known to Lie.In[27] Rieffel showed that given the linear Poisson bracket on h∗,a defor-mation quantization is provided by the convolution algebra structure on the simply connected Lie group H corresponding to h.In partic-ular,when h is a nilpotent Lie algebra,this is shown to be a strict deformation quantization.In this paper,we wish to generalize the above situation and to include twisted group convolution algebras into the framework of deformation quantization.Wefirst define a class of Poisson brackets on the dual vector space of a Lie algebra,which contains the linear Poisson bracket as a special case.These Poisson brackets can actually be realized as “central extensions”of the linear Poisson bracket.We then show that twisted group convolution algebras provide deformation quantizations of these Poisson brackets.We obtain strict deformation quantizations when the Lie algebra is nilpotent.In addition to its interest as a generalization of the deformation quantization problem into the non-linear situation,this result has a nice application to quantum groups.Quantum groups[13],[7]are usually obtained by suitably“deforming”ordinary Lie groups,and as suggested by Drinfeld[13],we expect to obtain quantum groups by deformation quantization of the so-called Poisson–Lie groups[19].In some cases, the compatible Poisson brackets on the Poisson–Lie groups are shown to be of our special type,in which case we can apply the result of this paper to obtain(strict)deformation quantizations of them.This enables us to construct some specific non-compact quantum groups.Not only have we actually been able to show[16]that some of the earlier known examples of non-compact quantum groups[28],[31], [11],[18]are obtained in this way,but we also obtain a new class of non-compact quantum groups[15].Although the method of construction may seem rather naive,our new example is shown to satisfy some interesting properties,including the“quasitriangular”property.We will discuss our construction of quantum groups in a separate paper. This paper is organized as follows.In thefirst section,we review the definitions of Poisson brackets and the formulation of(strict)de-formation quantization.We also include a discussion on twisted group algebras,which are the main objects of our study.In the second section, we define our special class of non-linear Poisson brackets,as motivated by the central extension of ordinary Lie brackets.We show in the third section that certain twisted group C∗–algebras provide strict defor-mation quantizations of these special Poisson brackets.We use some non-trivial results on twisted group C∗–algebras obtained by PackerDEFORMATION QUANTIZATION OF POISSON STRUCTURES3 and Raeburn[21,22].We restrict our study to the strict deforma-tion quantization case,but some indications for generalization are also briefly mentioned.The essential part of this article is from the author’s Ph.D.thesis at U.C.Berkeley.I would like to express here my deep gratitude to Professor Marc Rieffel,without whose constant encouragement,show of interest and numerous suggestions,this work would not have been made possible.1.PreliminariesLet M be a C∞manifold,and let C∞(M)be the algebra of complex-valued C∞functions on M.It is a commutative algebra under point-wise multiplication,and is equipped with an involution given by com-plex conjugation.Definition1.1.By a Poisson bracket on M,we mean a skew,bilinear map{,}:C∞(M)×C∞(M)→C∞(M)such that the following holds:•{,}defines a Lie algebra structure on C∞(M).(i.e.the bracket satisfies the Jacobi identity.)•(Leibniz rule):{f,gh}={f,g}h+g{f,h},for f,g,h∈C∞(M). We also require that the Poisson bracket be real,in the sense that {f∗,g∗}={f,g}∗.A manifold M equipped with such a bracket is called a Poisson manifold,and C∞(M)is a Poisson(*–)algebra. The deformation quantization will take place in C∞(M)(or to allow non-compact M,in C∞∞(M),which is the space of C∞functions van-ishing at infinity).The Poisson bracket on M gives the direction of the deformation.Let us formulate the following definition for deformation quantization,which is the“strict”deformation quantization proposed by Rieffel[26].Depending on the situations,we may also be inter-ested in some other subalgebras of C∞(M),on which the deformation takes place.So the definition is formulated at the level of an arbitrary (dense)∗–subalgebra A,for example the algebra of Schwartz functions. Definition1.2.Let M be a Poisson manifold as above.Let A be a dense∗–subalgebra(with respect to the C∗–norm ∞)of C∞(M), on which{,}is defined with values in A.By a strict deformation quantization of A in the direction of{,},we mean an open interval I of real numbers containing0,together with a triple(× ,∗ , )for each ∈I,of an associative product,an involution,and a C∗–norm (for× and∗ )on A,such that4BYUNG–JAY KAHNG1.For =0,the operations× ,∗ , are the original pointwiseproduct,involution(complex conjugation),and C∗–norm(i.e.sup-norm ∞)on A,respectively.2.The completed C∗-algebras A form a“continuousfield”of C∗–algebras(In particular,the map → f is continuous for any f∈A.).3.For any f,g∈A,f× g−g× fDEFORMATION QUANTIZATION OF POISSON STRUCTURES5 For convenience,let us denote byσr,r∈N the ordinary group cocycle on G defined byσr(x,y)=σ(x,y;r).Corresponding to the continuousfield of cocyclesσ:r→σr,we can define[25]the twisted convolution and involution on L1 G,C∞(N,A) .We define,forφ,ψ∈L1 G,C∞(N,A) :(φ∗σψ)(x;r)= Gφ(y;r)αy ψ(y−1x;r) σr(y,y−1x)dyandφ∗(x;r)=αx φ(x−1;r)∗ σr(x,x−1)∗∆G(x−1).We thus obtain a Banach∗–algebra.Let us denote this algebra byL1(G,N,A,σ),with the group actionαto be understood.We also de-fine C∗(G,N,A,σ),the enveloping C∗–algebra of L1(G,N,A,σ).There are also the notions of induced representations and regular representa-tions[5],[37],[25].So we may as well define the reduced C∗–algebraC∗r(G,N,A,σ).All these are more or less straightforward.Compare this definition with the definition given in[5],where the twisted group convolution algebra has been formulated via a single cocycle.Nevertheless,the present definition is no different from the usual one,since we can regardσalso as a single cocycle,taking values in UZM C∞(N,A) .The present formulation is useful when we study the continuity problem of thefields of C∗–algebras consisting of twisted group C∗–algebras,by varying the cocycles.Recall that the continuousfield property is essential in the definition of the strict deformation quantization(Definition1.2).Using the universal property of full C∗–algebras,and also taking ad-vantage of the property of the reduced C∗–algebras that one is able to work with their specific representations,Rieffel in[25]gave an answer to the problem of the continuity of thefield of C∗–algebras C∗(G,A,σr) r∈N, as follows.Here C∗(G,A,σr)is the twisted group C∗–algebra in the usual sense of[5].Theorem1.4.Let G,A,αbe understood as above.Letσbe a contin-uousfield over N ofα–cocycles on G.Then•Thefield C∗(G,A,σr) over N is upper semi-continuous.• C∗r(G,A,σr) over N is lower semi-continuous.•Thus,if each(G,A,α,σr)satisfies the“amenability condition”,i.e.C∗r(G,A,α,σr)=C∗(G,A,α,σr),then it follows that thefield of C∗–algebras C∗(G,A,σr) r∈N is continuous.For the proof of the theorem and the related questions,we will refer the reader to[25],and the references therein.Note that by replacing6BYUNG–JAY KAHNGA with C∞(N,A)and by introducing a new base space,we may even consider a continuousfield of the twisted group C∗–algebras given by the cocycles of continuousfield type(Definition1.3).The C∗–algebra C∗(G,N,A,σ)may be regarded as a C∗–algebra of “cross sections”of the continuousfield C∗(G,A,σr) r∈N.It is called the C∗–algebra of sections of a C∗–bundle by Packer and Raeburn[22] (Compare this terminology with Fell’s notion of“C∗–algebraic bun-dles”[14],which is considerably more general than is needed for our present purposes.).Actually in[22],the continuity problem of twisted group C∗–algebras allowing both the cocycle and the action to vary continuously has been studied in terms of the aforementioned notion of section C∗–algebra of a C∗–bundle.Taking a related viewpoint, Blanchard in[3]has recently developed a framework for a general con-tinuousfield of C∗–algebras in terms of“C∞(X)–algebras”,where X in our case is the locally compact base space N.A C∞(X)–algebra is a certain C∗–algebra having a C(X)module structure.See[3].We conclude this section by quoting(without proof)a couple of deep theorems of Packer and Raeburn[21,22]on the structure of twisted group C∗–algebras.We tried to keep Packer and Raeburn’s notation and terminology.Although some of them are different from our no-tation,they are clear enough to understand.For example,A×α,u G denotes the twisted group C∗–algebra(or“twisted crossed product”) C∗(G,A,α,u).All this and more can be found in[21,22].These the-orems will be used later in the proof of our Theorem3.4,which is our main result.Theorem1.5.([21])(Decomposition of twisted crossed products)Sup-pose that(A,G,α,u)is a separable twisted dynamical system and N is a closed normal subgroup of G.There exists a canonically determined twisted action(β,v)of G/N on A×α,u N such that:A×α,u G∼=(A×α,u N)×β,v G/N.The next theorem is about the continuity of afield of twisted group C∗–pare this with Theorem1.4,where we considered the continuity problem only when the twisting cocycle is varying.Mean-while,note in the theorem that G is assumed to be amenable(So by the“stabilization trick”of Packer and Raeburn[21],the amenability condition always holds for any quadruple(G,A,α,u).).Theorem1.6.([22])Suppose A is the C∗–algebra of sections of a sep-arable C∗–bundle over a locally compact space X,and(α,u)is a twisted action of an amenable locally compact group G on A such that each ideal I x={a∈A:a(x)=0}is invariant.Then for each x∈X,there is aDEFORMATION QUANTIZATION OF POISSON STRUCTURES 7natural twisted action α(x ),u (x )on the quotient A/I x ,and A ×α,u G is the C ∗–algebra of sections of a C ∗–bundle over X with fibers isomor-phic to (A/I x )×α(x ),u (x )G .2.The non-linear Poisson bracketLet us begin by trying to characterize the special Poisson brackets that will allow twisted group algebras to be deformation quantizations of them.Recall that the twisting of the convolution algebra structure in a twisted group algebra is given by group (2–)cocycles.Meanwhile any group cocycle for a locally compact group G having values in an abelian group N can be canonically associated with a central extension of G by N ,and actually all central extensions are essentially obtained in this way [14].Since it is known [27]that ordinary group convolution algebras can be regarded as deformation quantizations of linear Poisson brackets,the above observations suggest that twisted group algebras will provide deformation quantizations of certain Poisson brackets which are,in a loose sense,“central extensions”of linear Poisson brackets.Although we have to make clear what we mean by this last statement,this is the main motivation behind the definition of our special type of Poisson bracket formulated below.Let h be a (finite–dimensional)Lie algebra and let us denote by g =h ∗its dual vector space.As usual,we will denote the dual pairing between h and g by , .We will have to require later that h is a nilpotent or an exponential solvable Lie algebra because of some technical reasons to be discussed below,but for the time being we allow h to be a general Lie algebra.Recall [35]that we define the linear Poisson bracket on the dual vector space g =h ∗by{φ,ψ}lin (µ)= [dφ(µ),dψ(µ)],µ(2.1)where φ,ψ∈C ∞(g )and µ∈g .Here dφ(µ)and dψ(µ)has been naturally realized as elements in h .We wish to define a generalization of this Poisson bracket by allowing a suitable “perturbation”of the right-hand side of equation (2.1).This will be done via a certain Lie algebra 2–cocycle on h ,denoted by Ω,having values in C ∞(g ).That is,we will consider the Poisson brackets of the form:{φ,ψ}(µ)= [dφ(µ),dψ(µ)],µ +Ω dφ(µ),dψ(µ);µ .(2.2)As above,we regard dφ(µ)and dψ(µ)as elements in h .8BYUNG–JAY KAHNGCompare equation(2.2)with the definition of the linear Poisson bracket.In the linear Poisson bracket case,the Lie bracket takes val-ues in h,the elements of which can be regarded as(linear)functions contained in C∞(g),via the dual pairing.That is,the right-hand side of equation(2.1)can be viewed as the evaluation atµ∈g of a C∞–function,[X,Y]∈h⊆C∞(g),where X=dφ(µ)and Y=dψ(µ). In the“perturbed”case,the right-hand side of equation(2.2)may be viewed as the evaluation atµ∈g of a C∞–function,[X,Y]+Ω(X,Y)∈C∞(g),where X=dφ(µ)and Y=dψ(µ).So to make sense of the Poisson brackets of the type given by equation(2.2),we willfirst study the“perturbation”of the Lie bracket on h by a ter,we willfind some additional conditions for the cocycleΩsuch that the equation(2.2)indeed gives a well-defined Poisson bracket on g.Let V be a U(h)–module,possibly infinite dimensional.Consider a 2–cocycleΩfor h having values in V.It is a skew-symmetric,bilinear map from h×h into V such that dΩ=0(For more discussion on cohomology of Lie algebras,see the standard textbooks on the subject [6,§5],[17].).When V is further viewed as an abelian Lie algebra,the space h⊕V can be given a Lie algebra structure[4],[17]which becomes a central extension Lie algebra of h by V:(X,v),(Y,w) h⊕V= [X,Y],X·w−Y·v+Ω(X,Y)for X,Y∈h and v,w∈V.Here the dot denotes the module action.In particular,when V is assumed to be a trivial U(h)–module,we have: (X,v),(Y,w) h⊕V= [X,Y],Ω(X,Y) .(2.3)Let us slightly modify this“central extension”picture as follows,so that we are able to consider a Lie bracket on h+V,where we now allow h∩V=0in general.Clearly,h∩V is a subspace of h.However,since h is already equipped with its given Lie bracket and since V will be assumed to be an abelian Lie algebra,it is only reasonable to consider the case in which h∩V is an abelian subalgebra of h.For simplicity,we will further assume that h∩V is a central subalgebra of h,which means that V is a trivial U(h)–module.Let us denote this central subalgebra by z.Without loss of generality,we may assume that z is the center of h.In this case,we just replace V by an extended abelian Lie algebra, still denoted by V,satisfying h∩V=z.Let us look for a trivial U(h)–module V,which we will view as an abelian Lie algebra,such that h∩V=z is the center of h.Since we eventually want to define a V–valued cocycle for h,from which we construct a bracket operation on C∞(g),we also require that V is contained in C∞(g).So let us consider the subspace q=z⊥of g,andDEFORMATION QUANTIZATION OF POISSON STRUCTURES 9choose as our V the following:V =C ∞(g /q )⊆C ∞(g ).Here the functions in V =C ∞(g /q )have been realized as functions in C ∞(g ),by the “pull-back”using the natural projection p :g →g /q .Since any X ∈h can be regarded as a linear function contained in C ∞(g )via the dual pairing,we can see easily that X ∈h ∩V ⊆C ∞(g )if and only if X,µ+ν = X,µ for all µ∈g ,ν∈q .It follows imme-diately that h ∩V =z .On the other hand,consider the representation ad ∗.For any X ∈h and any µ∈g ,we have ad ∗−X (µ)=ν∈q ,since for any Y ∈z ,we have Y,ν = Y,ad ∗−X (µ) = [X,Y ],µ =0.It follows that ad ∗X (f )(µ)=f ad ∗−X (µ) =f (ν)=0,for any f ∈V =C ∞(g /q ).By the natural extension of ad ∗to U (h ),we can give V the trivial U (h )–module structure.Remark.When h has a trivial center,the space V will be just {0}.To avoid this problem,we could have considered C ∞(g )H ,the space of Ad ∗H –invariant C ∞functions on g .It is always nonempty (It contains the so-called Casimir elements [32].).It also satisfies h ∩C ∞(g )H =z and can be given the trivial U (h )–module structure.However,it does not satisfy the following property (Lemma 2.1),which we need later when we define our Poisson bracket.For this reason,we choose our V as it is defined above.At least for nilpotent h ,which is the case we are going to study most of the time,this is less of a problem since h has a non-trivial center.Lemma 2.1.Let V ⊆C ∞(g )be defined as above.Then for any func-tion χ∈V and for any µ∈g ,we have:dχ(µ)∈z .Proof.Since V ⊆C ∞(g ),it follows that dχ(µ)∈h .Recall that dχ(µ)defines a linear functional on g =h ∗by dχ(µ),ν =d dt t =0χ(µ)=0.Since ν∈q is arbitrary,we thus have:dχ(µ)∈z .10BYUNG–JAY KAHNGLet us now turn to the discussion of defining a(perturbed)bracket operation on h+V,which will enable us to formulate our special Poisson bracket on C∞(g).Since h and V are subspaces of h+V,there exists a (linear)surjective map,h→(h+V)/V,whose kernel is h∩V=z.We thus obtain a vector space isomorphism,in a canonical way,between (h+V)/V and h/z.The map from h+V onto h/z is a canonical one, which extends the canonical projection of h onto h/z.Therefore,it is reasonable to consider a cocycle for h/z having values in V(viewed as a trivial U(h/z)–module)and use it to define a bracket operation on h+V.LetΩbe such a cocycle for h/z.Remark.Note that in this setting,the cocycleΩcan naturally be iden-tified with a cocycle˜Ωfor h having values in V(considered as a trivial U(h)–module),satisfying the following“centrality condition”:˜Ω(Z,Y)=˜Ω(Y,Z)=0,(2.4)for any Z∈z and any Y∈h.In fact,we may define˜Ωas˜Ω(X,Y)=Ω(˙X,˙Y),where˙X denotes the image in h/z of X under the canonical projection.For this reason,we will from time to time use the same notation,Ω,to denote bothΩand˜Ω.By viewingΩas a cocycle for h,we can define,as in equation(2.3), a Lie bracket on h⊕V:(X,v),(Y,w) h⊕V= [X,Y],Ω(X,Y) .To define a bracket operation on h+V,consider the natural surjective map from h⊕V onto h+V,whose kernel is:δ={(Z,−Z):Z∈z}⊆h⊕V.Sinceδis clearly central with respect to the Lie bracket[,]h⊕V given above,it is an ideal.Therefore,h+V=(h⊕V)/δis a Lie algebra. The Lie bracket on it is given by:(2.5)[X+v,Y+w]h+V=[X,Y]+Ω(X,Y),X,Y∈h,v,w∈V which is the given Lie bracket on h plus a cocycle term.In this sense, equation(2.5)may be considered as a“perturbed Lie bracket”of the given Lie bracket on pare this with equation(2.2),where the linear Poisson bracket(given by the Lie bracket on h)is“perturbed”by a certain cocycleΩ.Using the observation given above as motivation,let us define more rigorously our Poisson bracket on g=h∗.This is,in fact,a“cocycle perturbation”of{,}lin on g.DEFORMATION QUANTIZATION OF POISSON STRUCTURES11 Theorem2.2.Let h be a Lie algebra with center z and let g=h∗be the dual vector space of h.Consider the vector space V=C∞(g/q)⊆C∞(g)as above,where q=z⊥.Let us give V the trivial U(h)–module structure.LetΩbe a Lie algebra2–cocycle for h having values in V, satisfying the centrality condition.That is,Ωis a skew-symmetric, bilinear map from h×h into V such that:Ω X,[Y,Z] +Ω Y,[Z,X] +Ω Z,[X,Y] =0,X,Y,Z∈h satisfying:Ω(Z,Y)=Ω(Y,Z)=0for Z∈z and any Y∈h.Then the bracket operation{,}Ω:C∞(g)×C∞(g)→C∞(g)defined by {φ,ψ}Ω(µ)= [dφ(µ),dψ(µ)],µ +Ω dφ(µ),dψ(µ);µ is a Poisson bracket on g.Remark.If we denote dφ(µ)and dψ(µ)by X and Y,as elements in h,the right-hand side of the definition of the Poisson bracket may be viewed as the evaluation atµ∈g of a C∞–function,[X,Y]+Ω(X,Y)∈h+V∈C∞(g).Note that this expression is just the Lie bracket on h+V defined earlier by equation(2.5).Proof.Since dφ(µ)and dψ(µ)can be naturally viewed as elements in h,it is easy to see that{,}Ωis indeed a map from C∞(g)×C∞(g) into C∞(g).The skew-symmetry and bilinearity are clear.To verify the Jacobi identity,consider the functionsφ1,φ2,φ3in C∞(g).We may write{φ2,φ3}Ωas:{φ2,φ3}Ω(µ)={φ2,φ3}lin(µ)+χ(µ),whereχis a function in V.We therefore have:d {φ2,φ3}Ω (µ)= dφ2(µ),dφ3(µ) +dχ(µ).Thefirst term in the right hand side is the differential of the linear Poisson bracket,which is rather well known[35].Moreover,sinceχ∈V,it follows from Lemma2.1that dχ(µ)∈z,which is“central”with respect to both[,]andΩ.We thus have:φ1,{φ2,φ3}Ω Ω(µ)= dφ1(µ),d({φ2,φ3}Ω)(µ) ,µ +Ω dφ1(µ),d({φ2,φ3}Ω)(µ);µ = dφ1(µ),[dφ2(µ),dφ3(µ)] ,µ +Ω dφ1(µ),[dφ2(µ),dφ3(µ)];µ , and similarly for φ2,{φ3,φ1}Ω Ωand φ3,{φ1,φ2}Ω Ω.So the Jacobi identity for{,}Ωfollows from that of the Lie bracket[,]and the cocycle identity forΩ.That is,φ1,{φ2,φ3}Ω Ω(µ)+ φ2,{φ3,φ1}Ω Ω(µ)+ φ3,{φ1,φ2}Ω Ω(µ)=0.12BYUNG–JAY KAHNGFinally,since d(φψ)=(dφ)ψ+φ(dψ)for anyφ,ψ∈C∞(g),the Leibniz rule for the bracket is also clear.Remark.This is the special type of Poisson bracket we will work with from now on.The linear Poisson bracket{,}lin on g=h∗is clearly of this type,since it corresponds to the case when the cocycleΩis trivial.Meanwhile whenΩis a scalar-valued cocycle,we obtain the so-called affine Poisson bracket[2],[30].Affine Poisson structures oc-cur naturally in the study of symplectic actions of Lie groups with general(not necessarily equivariant for the coadjoint action)moment mappings.The notion of affine Poisson brackets has generalization also to the groupoid level.See[12]or[36].Suppose we are given a Poisson bracket of our special type,{,}Ω, on g=h∗.To discuss its(strict)deformation quantization,it is useful to observe that{,}Ωcan be viewed as a“central extension”of the linear Poisson bracket on the dual vector space of the Lie algebra h/z. This actually follows from the fact that the Lie bracket[,]h+V given by equation(2.5)can be transferred to a Lie bracket on h/z⊕V,which turns out to be a central extension of the Lie bracket on h/z.Let us make this observation more precise.Consider the exact sequence of Lie algebras,0→zι→hρ→h/z→0such thatιandρare the injection and the quotient map,respectively. Let usfix a linear mapτ:h/z→h such thatρτ=id.In this case,the exactness implies that the mapω0:(x,y)→ι−1 [τ(x),τ(y)]−τ([x,y]h/z)(2.6)is well-defined from h/z×h/z into z,and it is actually a Lie algebra cocycle for h/z having values in z.See[4],[17].Then the Lie bracket on h can be written as follows:[X,Y]=τ [ρ(X),ρ(Y)]h/z +ι ω0(ρ(X),ρ(Y)) ,X,Y∈h.(2.7)Since we have been regarding the center z as a subalgebra of h such that h∩V=z,we may ignore the mapιand view z and its image in h or V as the same.Thenτis actually the map that determines the vector space isomorphism between h/z⊕V and h+V.Under this isomorphism and by using equation(2.7),the Lie bracket[,]h+V of equation(2.5)is transferred to a Lie bracket on h/z⊕V defined by:(2.8)(x,v),(y,w) h/z⊕V=[x,y]h/z+ω0(x,y)+Ω(x,y)=[x,y]h/z+ω(x,y).DEFORMATION QUANTIZATION OF POISSON STRUCTURES13 Hereω0is the cocycle defined in equation(2.6)and we regardedΩas a cocycle for h/z,as assured by an earlier remark.For convenience,we introduced a new(V–valued)cocycleωfor h/z,as a sum of the two cocyclesω0andΩ.Then it is clear that equation(2.8)defines a central extension of the Lie bracket[,]h/z,where the extension is given by the cocycleω.We can now define a Poisson bracket on g modeled after this central extension type Lie bracket.Theorem2.3.Let h be a Lie algebra with center z and let usfix the mapsρandτgiven above.Let g=h∗.Consider the vector space V⊆C∞(g)defined above and let us give V the trivial U(h/z)–module structure.Supposeωis a Lie algebra cocycle for h/z having values in V.Then the bracket operation{,}ω:C∞(g)×C∞(g)→C∞(g) defined by{φ,ψ}ω(µ)= τ([˙dφ(µ),˙dψ(µ)]h/z),µ +ω ˙dφ(µ),˙dψ(µ);µ is a Poisson bracket on g.Here˙X denotes the image of X under the canonical projectionρof h onto h/z.Proof.DefineΩ:h/z×h/z→V by(2.9)Ω(x,y)=ω(x,y)−ω0(x,y),whereω0is the cocycle for h/z defined in equation(2.6).Then the discussion in the previous paragraph implies that the bracket{,}ωis equivalent to the Poisson bracket{,}Ωgiven in Theorem2.2.There-fore,it is clearly a Poisson bracket on g.Although the present formulation depends on the choice of the map τand hence is not canonical,this Poisson bracket is,by construction, equivalent to the canonical Poisson bracket given in Theorem2.2.The relationship between them is given by equation(2.9).In particular,if we consider the cocycleω0of equation(2.6)in place ofω,so thatΩis trivial,we obtain the linear Poisson bracket{,}lin on g.There-fore,tofind a(strict)deformation quantization of our Poisson bracket {,}Ωof Theorem2.2,we may as well try tofind a(strict)deforma-tion quantization of the central extension type Poisson bracket{,}ω. This change in our point of view is useful when we work with specific examples,where the choice of coordinates are usually apparent.3.Twisted group C∗–algebras as deformationquantizationsAs we mentioned earlier,we expect that twisted group C∗–algebras will be deformation quantizations of the Poisson brackets of“central extension”type.These are in fact the special type of Poisson brackets14BYUNG–JAY KAHNGwe defined in the previous section.A more canonical description has been given in Theorem2.2,while an equivalent,“central extension”type description has been given in Theorem2.3.Let us from now on consider the Poisson bracket{,}ωon g=h∗, as defined in Theorem2.3.For convenience,we willfix the map τ:h/z→h and identify h/z with its imageτ(h/z)⊆h underτ. Tofind a deformation quantization of{,}ω,we will look for a group (2–)cocycle,σ,for the Lie group H/Z of h/z,corresponding to the Lie algebra cocycleω.Then we will form a twisted group C∗–algebra of H/Z withσ,which we will show below will give us a strict deformation quantization of C∞(g)in the direction of{,}ω.By the equivalence of the Poisson brackets{,}ωand{,}Ω,this may also be interpreted as giving a strict deformation quantization of C∞(g)in the direction of{,}Ω.This result will be a generalization of the result by Rieffel [27]saying that an ordinary group C∗–algebra C∗(H)provides a defor-mation quantization of C∞(h∗)in the direction of the linear Poisson bracket on h∗.Recall that the cocycleωprovides a Lie bracket on the space h/z⊕V. If we restrict this Lie bracket to h/z,we obtain the map[,]ω:h/z×h/z→h/z⊕V defined by:[x,y]ω=[x,y]h/z+ω(x,y).(3.1)For the time being,to make our book keeping simpler,let us denote by k and K the Lie algebra h/z and its Lie group H/Z.We now try to construct a group-like structure corresponding to[,]ω.From equation(3.1),we expect to obtain a cocycle extension of the Lie group K=H/Z via a certain group cocycle corresponding toω.As afirst step,let us consider the following Baker–Campbell–Hausdorffseries for k⊕V,ignoring the convergence problem for the moment.DefineS(X,Y)=X+Y+112[X,[X,Y]k⊕V]k⊕V+1S( X, Y)for=0in R.For =0,we let S(X,Y)=X+Y.Lemma3.1.Let ∈R befixed and let S be as above.Then we have, at least formally(ignoring the convergence problem),S X,S (Y,Z) =S S (X,Y),ZS (X,−X)=0,S (X,0)=S (0,X)=Xfor X,Y,Z∈k⊕V.。
