下载文档原格式
建筑工程学院英文School of Architecture and EngineeringIntroductionThe School of Architecture and Engineering is one of the leading institutions in the field of architectural studies and engineering education in our country. With a rich history and a commitment to academic excellence, our school is dedicated to fostering innovation, creativity, and technical expertise among our students. In this article, we will explore the various programs, facilities, and achievements of the School of Architecture and Engineering.Programs OfferedThe School of Architecture and Engineering offers a wide range of undergraduate and postgraduate programs to cater to the diverse interests and aspirations of our students. Our undergraduate programs include Bachelor of Architecture, Bachelor of Civil Engineering, and Bachelor of Environmental Engineering. These programs provide a solid foundation in theory, design, construction, and management, equipping students with the necessary skills to excel in the field.For those interested in furthering their studies, we offer a Master of Architecture program that focuses on advanced architectural design and urban planning. Additionally, we have Master of Civil Engineering and Master of Environmental Engineering programs that delve deeper into specialized areas within their respective fields. Furthermore, we have adoctoral program for students who aspire to conduct research in architecture and engineering.Faculty and ResearchThe School of Architecture and Engineering boasts a highly qualified and dedicated faculty, comprising renowned professors, experienced practitioners, and industry experts. Our faculty members are actively engaged in research and have contributed significantly to the advancement of knowledge in their respective domains. Students benefit from their expertise and guidance, leading to a well-rounded education that combines theory and practical application.The research activities within the school cover a broad spectrum of topics, including sustainable architecture, structural engineering, urban design, and environmental planning. Through collaborations with industry partners and government agencies, our faculty and students actively participate in research projects that aim to address real-world challenges and contribute to the development of sustainable and resilient built environments.Infrastructure and FacilitiesThe School of Architecture and Engineering is equipped with state-of-the-art facilities to support teaching, learning, and research activities. Our campus houses design studios, laboratories, workshop spaces, and computer labs, providing students with ample opportunities to engage in hands-on learning experiences. Additionally, we have specialized software and equipment that facilitate the exploration of digital design, building information modeling, and structural analysis.To encourage interdisciplinary collaborations, our school also offers shared spaces and collaborative areas where students and faculty from different disciplines can come together and exchange ideas. These spaces promote a vibrant academic environment and foster a sense of community among students and faculty members.Industry Connections and Experiential LearningRecognizing the importance of practical experience in the field of architecture and engineering, the School of Architecture and Engineering has established strong connections with the industry. We collaborate with renowned architectural firms, construction companies, and engineering consultancies to provide our students with internships, site visits, and opportunities to work on real-world projects. These experiences allow students to apply their theoretical knowledge in a practical setting, enhancing their skills and preparing them for the challenges of the profession.ConclusionThe School of Architecture and Engineering is committed to delivering a comprehensive and contemporary education in architecture and engineering. With a focus on academic excellence, research innovation, and industry connections, our school prepares students to become competent professionals who can contribute to the sustainable development of our built environment. We take pride in our accomplishments and look forward to nurturing future leaders in the field of architecture and engineering.(Note: The word count currently stands at 574 words. If a longer article is required, please provide additional details or specific areas of focus to continue writing.)。
建筑英语unit6Unit 6 Chinese ArchitectureText ANew words:●Axis n. 轴，轴线—— an imaginary line through the centre of an object, around which theobject turn.●Position v. 安放，安置——to put sb/sth in a particular position.●Ethnical a. 种族的——connected with or belonging to a nation, race or tribe that shares acultural tradition.●Neolithic a. 新石器时代的——of the later part of the STONE AGE.●Antisepsis n. 防腐（法），抗菌（法）——a substance that helps to prevent infection in woodsby killing bacteria.●Feudal a. 封建（制度）的——connected with or similar to feudalism.●Deem v. 认为——to have a particular opinion about sth.●Phoenix n. 凤凰——a mythical bird, the only one of its kind, that after living for five or sixcenturies in the Arabian desert, burnt itself on a funeral pyre and rose from the ashes with renewed youth to live through another cycle.●Foil n. 箔，金属薄片——metal made into very thin sheets that is used for covering orwrapping things.●Hallmark n. 标志，特点——a feature or quality that is typicalof sb/sth.●Cluster n. 串，丛，簇——a group of things of the same type that grow or appear closetogether.●Buddhist a. 佛教的——an Asian religion based on the teaching of Buddha.●Ward off 避开，挡开——to protect or defend yourself against danger ,illness, attack,etc.●Mythical a. 神话的——that does not exist or is not true.(connected with ancient MYTHS.)●Vegetation n. 植被——plants in general, especially the plants that are found in a particulararea or environment.●Trinity n. 三位一体，三合一——（in Christianity）the union of Father, Son andHOL Y SPIRIT as one God.●Repulse v. 击退，驱逐——to fight sb who is attacking you and drive them away.●Cohabitation n. 共生，共同存在——to live together.●Occult a. 超自然的，神秘的——connected with magic powers and things that cannot beexplained by reason or science.Architectural terms:●Bonding n. 接，接合——the process of forming a relationship with sth.●Joist n. 托梁——a long thick piece of wood or metal that is used to support a floor or ceilingin a building.●Overhanging a. 突出的——to stick out over and above sth else.●Tier building 多层建筑——a row or layer of sth that has several rows or layers placed oneabove the building.●Bracket n. 托架，支架，斗拱——a piece of wood, metal or plastic fixed to the wall tosupport a shelf, lamp, etc.●Glazed tiling 釉面砖，琉璃瓦——a thin clear liquid put on clay objects before they arefinished, to give them a hard shiny surface.●Spirit gate 神门Proper names:●Tang Dynasty 唐朝——(618 -907)was an imperial dynasty of China preceded by the SuiDynasty and followed by the Five Dynasties and Ten Kingdoms Period.●Vietnam 越南——is the easternmost country on the Indochina Peninsula in Southeast Asia.It is bordered by People's Republic of China (PRC) to the north.●Shang Dynasty 商朝——(1766 -1122 BC)was according to traditional sources the secondChinese dynasty, after the Xia Dynasty. They ruled in the northeastern regions of the area known as ―China proper‖ in the Yellow River valley.●Zhou Dynasty 周朝——(1046–256 BC) followed the Shang Dynasty and was followed bythe Qin Dynasty in China. The Zhou dynasty lasted longer than any other dynasty in Chinese history.●Han Dynasty 汉朝——(206 –220 BC) was the second imperial dynasty of China, precededby the Qin Dynasty. It was founded by the peasant rebel leader Liu Bang, known posthumously as Emperor Gaozu of Han.Text BNew words:●Embrace v. 包含——to comprise or include as an integral part.●Veritable a. 真正的，名副其实的——genuine or true。
Unit 3TEXT ANew wordscampusn.[C, U] the land and buildings of a university or college （大学或学院的）校园All freshman students live on campus. When they are in their second year at college, they may live off campus. 所有大学一年级的学生都住在校园里。
大学二年级时，他们可以住在校外。
transformvt.completely change the appearance, form, or character of sth. or sb., esp. in a way that improves it 使改观；使变形；使转化The president of the university said that they were trying their best to transform their university into a top school in the country. 这个大学的校长表示，他们正竭尽全力把他们的学校建设成为全国的一流大学。
fleetn.[C] a group of vehicles, planes, boats, or trains, esp. when they are owned by one organization or person 车队；机群；船队Survivors were taken to a hospital in a fleet of ambulances. 幸存者被救护车队送往医院。
FedEx has a fleet of trucks. 联邦快递有卡车车队。
typicala.like most things of the same type 典型的；有代表性的Notice the sentences in the text that are relatively long, which is typical of a news report. 注意这篇文章中的句子比较长，这在新闻报道中是很典型的。
U1(选修六)1.Abstract1)Adj.深奥的，抽象的Astronomy is an abstract subject. 天文学是一门深奥的学科。
The word “honesty〞is an abstract noun.Beauty is abstract but a house is not .美是抽象的，房子是具体的。
2〕V.○1“提炼〞“抽取〞The workers are abstracting metal from ore.工人们正在由矿砂提炼金属。
Rubber is abstractedfrom trees.橡胶是从树木提取的。
Salt can be abstracted from sea water.盐是从海水中提取出来的。
○2“转移〔注意〕等distract one’s attention from sth从……上转移开某人的注意力Nothing can distract his attention from his work.○3“概括，写摘要〞He is abstracting a story for a book review.他在为一篇书评撰写故事摘要。
3〕n.an abstract of a lecture一个演讲的摘要2.Would you rather have Chinese or Western-style paintings in your home?would rather do sth情愿做….would rather sb did sth情愿sb做…情愿做….而不愿意做…：would rather do sth than do sth= would do sth rather than do sth= prefer to do sth rather than do sth= prefer doing sth to doing sthI would rather stay at home today. 我今天宁愿待在家里。
New wordsUnit 6 TEXT Anumerousa.many 许多的；很多的The library has numerous books, more than I have ever expected. 这个图书馆拥有大量的图书，比我预想的要多得多。
reliablea.able to be trusted or depended on 可信赖的；可靠的A reliable employee does his/her job with minimal error. 一个可靠的员工工作起来错误最少。
contrastvt.compare two things, ideas, people, etc. to show how different they are from each other 使成对比；使成对照In her essay, the author contrasts the present economic crisis with the one 10 years ago. 作者在文中就当前的经济危机和十年前的经济危机进行了对比。
vi.(of two things) be different from each other, often in a noticeable or interesting way 形成对比Her dark hair contrasted sharply with her pale silk gown. 她的黑头发和她的浅色丝绸礼服形成了强烈的对比。
n.[C, U] a difference between people, ideas, situations, things, etc. that are being compared 差异；差别The book presents a very interesting contrast between life now and life 100 years ago. 这本书把现在的生活和100 年之前的生活进行了十分有趣的对比。
选修六u n i t1知识点。
1.Abstract1)Adj.深奥的，抽象的Astronomy is an abstract subject. 天文学是一门深奥的学科。
Beauty is abstract but a house is not .美是抽象的，房子是具体的。
2）V.○1“提炼”“抽取”Rubber is abstracted from trees.橡胶是从树木提取的。
Salt can be abstracted from sea water.盐是从海水中提取出来的。
②“概括，写摘要”He is abstracting a story for a book review.他在为一篇书评撰写故事摘要。
3）n.摘要an abstract of a lecture一个演讲的摘要2.would rather do sth 情愿做….would rather sb did sth 情愿sb做…I would rather stay at home today. 我今天宁愿待在家里。
I would rather you came here tomorrow. 我宁愿你明天来。
情愿做….而不愿意做…：would rather do sth than do sth= would do sth rather than do sth= prefer to do sth rather than do sth= prefer doing sth to doing sthI would rather go with you than stay here.I would go with you rather than stay here.I prefer to go with you rather than stay here.I prefer going with you to staying here.3.faith n. 信任,信仰break one's faith with sb. 对某人不守信用keep faith with 忠于信仰; 守信I kept faith with him.我信守了对他的诺言。
高中英语新课标选修6unit4高中英语新课标选修6的Unit 4通常围绕一个特定的主题展开，例如文化、科技、环境等。
由于我没有具体的教材内容，我将提供一个通用的Unit 4教学内容框架，以供参考。
Unit 4: Exploring Cultural DiversitySection 1: Warm-up and Vocabulary- Begin the class with a brief discussion about the importance of cultural diversity.- Introduce new vocabulary related to the theme, such as "heritage," "tradition," "cuisine," "folklore," and "customs."Section 2: Reading Comprehension- Present a reading passage that explores different aspects of cultural diversity, such as festivals, art, and music from various countries.- After reading, ask students to identify the main ideas and supporting details.- Discuss any unfamiliar expressions or idiomatic language used in the text.Section 3: Grammar Focus- Introduce a grammar point relevant to the theme, such ascomparative and superlative adjectives, which can be used to describe cultural differences.- Provide examples and practice exercises for students to complete.Section 4: Listening Practice- Play an audio recording of a conversation or a lecture about cultural exchange programs or experiences.- Have students listen for specific information and answer comprehension questions.Section 5: Speaking Activity- Organize a role-play activity where students take on the roles of cultural ambassadors from different countries, sharing information about their "country's" culture with the class.- Encourage the use of the new vocabulary and grammar structures learned in the previous sections.Section 6: Writing Task- Assign a writing task where students describe a cultural event or tradition from their own or another culture.- Provide a template or outline to guide their writing, focusing on organization and the use of descriptive language.Section 7: Cultural Appreciation- Conclude the unit with a class activity that involvesstudents sharing and appreciating various cultural elements, such as food, music, or clothing.- Encourage reflection on the importance of respecting and understanding different cultures.Section 8: Assessment- Administer a quiz or test to assess students' understanding of the vocabulary, grammar, and reading comprehension skills covered in the unit.Section 9: Homework- Assign homework that reinforces the learning objectives of the unit, such as further reading on a cultural topic, additional grammar exercises, or a short essay on cultural diversity.This framework is designed to be adaptable to the specific content and requirements of the actual textbook or curriculum being used. Adjustments can be made based on the needs and interests of the students.。
a rXiv:funct-a n /9724v111F eb1997PSEUDODIFFERENTIAL OPERATORS ON DIFFERENTIAL GROUPOIDS VICTOR NISTOR,ALAN WEINSTEIN,AND PING XU Abstract.We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners.We show that this construction encompasses many examples.The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid,in the sense of non-commutative geometry.Symbol calculus for our algebra lies in the Poisson algebra of functions on the dual of the Lie algebroid of the groupoid.As applications,we give a new proof of the Poincar´e -Birkhoff-Witt theorem for Lie algebroids and a concrete quantization of the Lie-Poisson structure on the dual A ∗of a Lie algebroid.Contents Introduction 11.Preliminaries 32.Main definition 63.Differential operators and quantization 134.Examples 185.Distribution kernels 216.The action on sections of E 26References 28Introduction Certain important applications of pseudodifferential operators require variantsof the original definition.Among the many examples one can find in the literature are regular or adiabatic families of pseudodifferential operators [2,40]and pseudo-differential operators along the leaves of foliations [6,8,27,28],on coverings [9,29]or on certain singular spaces [21,22,26,25].Since these classes of operators share many common features,it is natural to ask whether they can be treated in a unified way.In this paper we shall suggest an answer to this question.For any “almost differential”groupoid (a class which allows manifolds with corners),we construct an algebra of pseudodifferential operators.We then show that our construction recovers (almost)all the classes described above (for operators on manifolds with boundary our algebra is slightly smaller thanthe2V.NISTOR,A.WEINSTEIN,AND PING XUone defined in[21]).We expect our results to have applications to analysis on singular spaces,not only manifolds with corners.Our construction and results owe a great deal to the previous work of several authors,especially Connes[6]and Melrose[21,23,20].A hint of the direction we take was given at the end of[37].The basic idea of our construction is to consider families of pseudodifferential operators along thefibers of the domain(or source) map of the groupoid.More precisely,for any almost differentiable groupoid(see Definition3)we consider thefibers G x=d−1(x)of the domain map d consisting of all arrows with domain x.It follows from the definition that thesefibers are smooth manifolds(without corners).The calculus of pseudodifferential operators on smooth manifolds is well understood and by now a classical subject,see for example[14]. We shall consider differentiable families of pseudodifferential operators P x on the smooth manifolds G x.Right translation by g∈G defines an isomorphism G x≡G y where x is the domain of g and y is the range of g.We say that the family P x is invariant if P x transforms to P y under the diffeomorphisms above(for all g). The algebraΨ∞(G)of pseudodifferential operators on G that we shall consider will consist of invariant differentiable families of operators P x as explained above (the actual definition also involves a technical condition on the support of these operators).See Definition7for details.The relation with the work of Melrose relies on an alternative description of our algebra as an algebra of distributions on G with suitable properties(compactly supported,and conormal with singular support contained in the set of units).This is contained in Theorem7.The difference between our theory and Melrose’s lies in the fact that he considers a compactification of G as a manifold with corners,and his distributions are allowed to extend to the compactification,with precise behavior at the boundary.This is useful for the analysis of these operators.In contrast,our work is purely algebraic(or geometric,depending on whether one considers Lie algebroids as part of geometry or algebra).We now review the contents of the sections of this paper.In thefirst section we recall the definitions of a groupoid,Lie algebroid and,the less known definition of a local Lie groupoid.We extend the definition of a Lie groupoid to include manifolds with corners.These groupoids are called almost differentiable groupoids. The second section contains the definition of a pseudodifferential operator on a groupoid(really a family of pseudodifferential operators,as explained above)and the proof that they form an algebra,if a support condition is included.We also extend this definition to include local Lie groupoids.This is useful in the third section where we use this to give a new proof of the Poincar´e-Birkhoff-Witt theorem for Lie algebroids.In the process of proving this theorem we also exemplify our definition of pseudodifferential operators on an almost differentiable groupoid by describing the differential operators in this class.As an application we give an explicit construction of a deformation quantization of the Lie-Poisson structure on A∗,the dual of Lie algebroid A.The section entitled“Examples”contains just what the title suggests:for many particular examples of groupoids G we explicitly describe the algebraΨ∞(G)of pseudodifferential operators on G.This recovers classes of operators that were previously defined using ad hoc constructions.Our definition is often not only more general,but also simpler.This is the case for operators along the leaves of foliations[8,27]or adiabatic families of operators.PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS3 Since one of our main themes is that the Lie algebras of vectorfields which are central in[24]are in fact the spaces of sections of Lie algebroids,we describe these Lie algebroids explicitly in each of our examples.In the sixth section of the paper,we describe the convolution kernels(called reduced kernels)of operators inΨ∞(G).Then we extend to our setting some fun-damental results on principal symbols,by reducing to the classical results.This makes our proofs short(and easy).Finally,the last section treats the action of Ψ∞(G)on functions on the units of G,and a few related topics.Thefirst author would like to thank Richard Melrose for several useful conver-sations.1.PreliminariesIn the following we allow manifolds to have corners.Thus by“manifold”we shall mean a C∞manifold,possibly with corners,and by a“smooth manifold”we shall mean a manifold without corners.By definition,if M is a manifold with corners then every point p∈M has coordinate neighborhoods diffeomorphic to [0,∞)k×R n−k.The transition functions between such coordinate neighborhoods must be smooth everywhere(including on the boundary).We shall use the following definition of submersions between manifolds(with corners).Definition1.A submersion between two manifolds with corners M and N is a differentiable map f:M→N such that d f x:T x M→T f(x)N is onto for any x∈M,and if d f x(v)is an inward pointing tangent vector to N,then v is an inward pointing tangent vector to M.The reason for introducing the definition above is that for any submersion f: M→N,the set M y=f−1(y),y∈N is a smooth manifold,just as for submersions of smooth manifolds.We shall study groupoids endowed with various structures.([32]is a general reference for some of what follows.)We recallfirst that a small category is a category whose class of morphisms is a set.The class of objects of a small category is then a set as well.Definition2.A groupoid is a small category G in which every morphism is in-vertible.This is the shortest but least explicit definition.We are going to make this definition more explicit in cases of interest.The set of objects,or units,of G will be denoted byM=G(0)=Ob(G).The set of morphisms,or arrows,of G will be denoted byG(1)=Mor(G).We shall sometimes write G instead of G(1)by abuse of notation.For example, when we consider a space of functions on G,we actually mean a space of functions on G(1).We will denote by d(g)[respectively r(g)]the domain[respectively,the range]of the morphism g:d(g)→r(g).We thus obtain functions(1)d,r:G(1)−→G(0)4V.NISTOR,A.WEINSTEIN,AND PING XUthat will play an important role bellow.The multiplication operator µ:(g,h )→µ(g,h )=gh is defined on the set of composable pairs of arrows G (2):µ:G (2)=G (1)×M G (1):={(g,h ):d (g )=r (h )}−→G (1).(2)The inversion operation is a bijection ι:g →g −1of G (1).Denoting by u (x )the identity morphism of the object x ∈M =G (0),we obtain an inclusion of G (0)into G (1).We see that a groupoid G is completely determined by the spaces G (0)and G (1)and the structural morphisms d,r,µ,u,ι.We sometimes write G =(G (0),G (1),d,r,µ,u,ι).The structural maps satisfy the following properties:(i)r (gh )=r (g ),d (gh )=d (h )for any pair (g,h )∈G (2),and the partially defined multiplication µis associative.(ii)d (u (x ))=r (u (x ))=x ,∀x ∈G (0),u (r (g ))g =g and gu (d (g ))=g ,∀g ∈G (1)and u :G (0)→G (1)is one-to-one.(iii)r (g −1)=d (g ),d (g −1)=r (g ),gg −1=u (r (g ))and g −1g =u (d (g )).Definition 3.An almost differentiable groupoid G =(G (0),G (1),d,r,µ,u,ι)is a groupoid such that G (0)and G (1)are manifolds with corners,the structural maps d,r,µ,u,ιare differentiable,and the domain map d is a submersion.We observe that ιis a diffeomorphism and hence d is a submersion if and only if r =d ◦ιis a submersion.Also,it follows from the definition that each fiber G x =d −1(x )⊂G (1)is a smooth manifold whose dimension n is constant on each connected component of G (0).The ´e tale groupoids considered in [5]are extreme examples of differentiable groupoids (corresponding to dim G x =0).If G (0)is smooth (i.e.if it has no corners)then G (1)is also smooth and G becomes what is known as a differentiable,or Lie groupoid.1We now introduce a few important geometric objects associated to an almost differentiable groupoid.The vertical tangent bundle (along the fibers of d )of an almost differentiable groupoid G isT d G =ker d ∗=x ∈G (0)T G x ⊂T G (1).(3)Its restriction A (G )=T d GG (0)to the set of units is the Lie algebroid of G [19,30].We denote by T ∗d G the dual of T d G and by A ∗(G )the dual of A (G ).In addition tothese bundles we shall also consider the bundle Ωλd of λ-densities along the fibers of d .If the fibers of d have dimension n then Ωλd =|Λn T ∗d G|λ=∪x Ωλ(G x ).By invariance these bundles can be obtained as pull-backs of bundles on G (0).For example T d G =r ∗(A (G ))and Ωλd =r ∗(D λ)where D λdenotes Ωλd |G (0).If E is a (smooth complex)vector bundle on the set of units G (0)then the pull-back bundle r ∗(E )on G will have right invariant connections obtained as follows.A connection ∇on E lifts to a connection on r ∗(E ).Its restriction to any fiber G x defines a linear connection in the usual sense,which is denoted by ∇x .It is easy to see that these connections are right invariant in the sense thatR ∗g ∇x =∇y ,∀g ∈G such that r (g )=x and d (g )=y.(4)The bundles considered above will thus have invariant connections.PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS5 The bundle A(G),called the Lie algebroid of G,plays in the theory of almost differentiable groupoids the rˆo le Lie algebras play in the theory of Lie groups.We recall for the benefit of the reader the definition of a Lie algebroid[30].Definition4.A Lie algebroid A over a manifold M is a vector bundle A over M together with a Lie algebra structure on the spaceΓ(A)of smooth sections of A,and a bundle mapρ:A→T P,extended to a map between sections of these bundles, such that(i)ρ([X,Y])=[ρ(X),ρ(Y)];and(ii)[X,fY]=f[X,Y]+(ρ(X)f)Yfor any smooth sections X and Y of A and any smooth function f on M.Note that we allow the base M in the definition above to be a manifold with corners.If G is an almost differentiable groupoid then A(G)will naturally have the struc-ture of a Lie algebroid[19].Let us recall how this structure is defined(the original definition easily extends to include manifolds with corners).Clearly A(G)is a vec-tor bundle.The right translation by an arrow g∈G defines a diffeomorphism R g:G r(g)∋g′→g′g∈G d(g).This allows us to talk about right invariant dif-ferential geometric quantities as long as they are completely determined by their restriction to all submanifolds G x.This is true of functions and d–vertical vector fields,and this is all that is needed in order to define the Lie algebroid structure on A(G).The sections of A(G)are in one-to-one correspondence with vectorfields X on G that are d–vertical,in the sense that d∗(X(g))=0,and right invariant.The condition d∗(X(g))=0means that X is tangent to the submanifolds G x,thefibers of d.The Lie bracket[X,Y]of two d–vertical right–invariant vectorfields X and Y will also be d–vertical and right–invariant,and hence the Lie bracket induces a Lie algebra structure on the sections of A(G).To define the action of the sections of A(G)on functions on G(0),observe that the right invariance property makes sense also for functions on G and that C∞(G(0))may be identified with the subspace of right–invariant functions on G.If X is a right–invariant vectorfield on G and f is a right–invariant function on G then X(f)will still be a right invariant function. This identifies the action ofΓ(A(G))on functions on G(0).Not every Lie algebroid is the Lie algebroid of a Lie groupoid(see[1]for an example).However,every Lie algebroid is associated to a local Lie groupoid[31]. The definition of a local Lie(or more generally,almost differentiable)groupoid [10]is obtained by relaxing the condition that the multiplicationµbe everywhere defined on G(2)(see Equation(2)),and replacing it by the condition thatµbe defined in a neighborhood U of the set of units.Definition5(van Est).An almost differentiable local groupoid L=(L(0),L(1))is a pair of manifolds with corners together with structural morphisms d,r:L(1)→L(0),ι:L(1)→L(1),u:L(0)→L(1)andµ:U→L(1),where U is a neighborhood of(u×u)(L(0))={(u(x),u(x))}in L(2)={(g,h),d(g)=r(h)}⊂L(1)×L(1). The structural morphisms are required to be differentiable maps such that d is a submersion,u is an embedding,and to satisfy the following properties:(i)The products u(d(g))g,gu(r(g)),gg−1and g−1g are defined and coincide with,respectively,g,g,u(r(g))and u(d(g));where we denoted g−1=ι(g)as usual.(ii)If gh is defined,then h−1g−1is defined and equal to(gh)−1.(iii)(Local associativity)If gg′,g′g′′and(gg′)g′′are defined then g(g′g′′)is also defined and equal to(gg′)g′′.6V.NISTOR,A.WEINSTEIN,AND PING XUThe set U is the set of arrows for which the product gh=µ(g,h)is defined.We see that the only difference between a groupoid and a local groupoid L is the fact that the condition d(g)=r(h)is necessary for the product gh=µ(g,h) to be defined,but not sufficient in general.The product is defined as soon as the arrows g and h are“small enough”.A consequence of this definition is that the right multiplication by an arrow g∈L(1)defines only a diffeomorphismU g−1∋g′→g′g∈U g(5)of an open(and possibly empty)subset U g−1of L y,y=r(g)to an open subset U g⊂L x,x=d(g).This will not affect the considerations above,however,so we can associate a Lie algebroid A(L)to any almost differentiable local groupoid L.In the following,when considering groupoids,we shall sometimes refer to them as global groupoids,in order to stress the difference between groupoids and local groupoids.2.Main definitionConsider a complex vector bundle E on the space of units G(0)of an almost differentiable groupoid G.Denote by r∗(E)its pull-back to G(1).Right translations on G define linear isomorphismsU g:C∞(G d(g),r∗(E))→C∞(G r(g),r∗(E))(6)(U g f)(g′)=f(g′g)∈(r∗E)g′which makes sense because(r∗E)g′=(r∗E)g′g=E r(g′).If G is merely a local groupoid then(6)is replaced by the isomorphismsU g:C∞(U g,r∗(E))→C∞(U g−1,r∗(E))(7)defined for the open subsets U g⊂G d(g)and U g−1⊂G r(g)defined in(5).Let B⊂R n be an open subset.Define the space S m(B×R n)of symbols on the bundle B×R n→B as in[14]to be the set of smooth functions a:B×R n→C such that|∂αy∂βξa(y,ξ)|≤C K,α,β(1+|ξ|)m−|β|(8)for any compact set K⊂B and any multiindicesαandβ.An element of one of our spaces S m should properly be said to have“order less than or equal to m”; however,by abuse of language we will say that it has“order m”.A symbol a∈S m(B×R n)is called classical if it has an asymptotic expansion as an infinite sum of homogeneous symbols a∼ ∞k=0a m−k,a l homogeneous of degree l:a l(y,tξ)=t l a l(y,ξ)if ξ ≥1and t≥1.(“Asymptotic expansion”is used here in the sense that a− N−1k=0a m−k belongs to S m−N(B×R n).)The space of classical symbols will be denoted by S m cl(B×R n).We shall be working exclusively with classical symbols in this paper.This definition immediately extends to give spaces S m cl(E;F)of symbols on E with values in F,whereπ:E→B and F→B are smooth euclidian vector bundles. These spaces,which are independent of the metrics used in their definition,are sometimes denoted S m cl(E;π∗(F)).Taking E=B×R n and F=C one recovers S m cl(B×R n)=S m cl(B×R n;C).A pseudodifferential operator P onB is a linear map P:C∞c(B)→C∞(B) that is locally of the form P=a(y,D y)plus a regularizing operator,where for anyPSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS7 complex valued symbol a on T∗W=W×R n,W an open subset of R n,one defines a(y,D y):C∞c(W)→C∞(W)bya(y,D y)u(y)=(2π)−n R n e iy·ξa(y,ξ)ˆu(ξ)dξ.(9)Recall that an operator T:C∞c(U)→C∞(V)is called regularizing if and only if it has a smooth distribution(or Schwartz)kernel.This happens if and only if T is pseudodifferential of order−∞.The class of a in S m cl(T∗W)/S m−1cl (T∗W)does not depend on any choices;thecollection of all these classes,for all coordinate neighborhoods W,patches togetherto define a classσm(P)∈S m cl(T∗W)/S m−1cl (T∗W)which is called the principal sym-bol of P.If the operator P acts on sections of a vector bundle E,then the principalsymbolσm(P)will belong to S m cl(T∗B;End(E))/S m−1cl (T∗B;End(E)).See[14]formore details on all these constructions.We shall sometimes refer to pseudodifferential operators acting on a smooth manifold as ordinary pseudodifferential operators,in order to distinguish them from pseudodifferential operators on groupoids,a class of operators which we now define (and which are really families of ordinary pseudodifferential operators).Throughout this paper,we shall denote by(P x,x∈G(0))a family of order m pseudodifferential operators P x,acting on the spaces C∞c(G x,r∗(E))for some vector bundle E over G(0).Operators between sections of two different vector bundles E1 and E2are obtained by considering E=E1⊕E2.Definition6.A family(P x,x∈G(0))as above is called differentiable if for any open set V⊂G,diffeomorphic through afiber preserving diffeomorphism to d(V)×W,for some open subset W⊂R n,and anyφ∈C∞c(V),we canfind a∈S m cl(d(V)×T∗V;End(E))such thatφP xφcorresponds to a(x,y,D y)under the diffeomorphism G x∩V≃W,for each x∈d(V).Afiber preserving diffeomorphism is a diffeomorphismψ:d(V)×W→V satisfying d(ψ(x,w))=x.Thus we require that the operators P x be given in local coordinates by symbols a x that depend smoothly on all variables,in particular on x∈G(0).Definition7.An order m invariant pseudodifferential operator P on an almost differentiable groupoid G,acting on sections of the vector bundle E,is a differ-entiable family(P x,x∈G(0))of order m classical pseudodifferential operators P x acting on C∞c(G x,r∗(E))and satisfyingP r(g)U g=U g P d(g)(invariance)(10)for any g∈G(1),where U g is as in(6).Replacing the coefficient bundle E by E⊗Dλand using the isomorphismΩλd≃r∗(Dλ),we obtain operators acting on sections of density bundles.Note that P can generally not be considered as a single pseudodifferential operator on G(1). This is because a family of pseudodifferential operators on a smooth manifold M, parametrized by a smooth manifold B,is not a pseudodifferential operator on the product M×B,although it acts naturally on C∞c(M×B).(See[2]or[14],page 94.)Recall[13]that distributions on a manifold Y with coefficients in the bundle E0 are continuous linear maps C∞c(Y,E′0⊗Ω)→C,where E′0is the dual bundle to E08V.NISTOR,A.WEINSTEIN,AND PING XUandΩ=Ω(Y)is the space of1-densities on Y.The collection of all distributions on Y with coefficients in the(finite dimensional complex vector)bundle E0is denoted C−∞(Y;E0).If P=(P x,x∈G(0))is a family of pseudodifferential operators acting on G x denote by k x the distribution kernel of P x(11)k x∈C−∞(G x×G x;r∗1(E)⊗r∗2(E)′⊗Ω2).HereΩ2is the pull-back of the bundle of vertical densitiesΩd on G x to G x×G x via the second projection.These distribution kernels are obtained using Schwartz’kernel theorem.We define the support of the operator P to besupp(P)=PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS9 Proof.If P consists of regularizing operators thenP f(g)= G x k x(g,h)f(h),where x=d(g).Lemma1implies that the formula above for P f involves only the integration of smooth(uniformly in g)compactly supported sections,and hence we can exchange integration and derivation to obtain the smoothness of P f.This proves(i)in case P consists of regularizing operators.The proof of(ii)if both P and Q consist of regularizing operators follows the same reasoning.We prove now(i)for P arbitrary.Fix g∈G x and V a neighborhood of gfiber preserving diffeomorphic to d(V)×W for some open convex subset W in R n,0∈W, such that(x,0)maps to g.Replacing P x by P x−R x for a smooth regularizing family R x we can assume that the distribution kernels k x of P x satisfyp−11(d(V)×W/4)∩∪supp(k′x)⊂(d(V)×W/2)×(d(V)×3W/4)where k′x are the distribution kernels of Q x.The support estimates above for P and Q show that the P y Q y for y∈d(V)are the compositions of smooth families of pseudodifferential operators acting on W⊂R n.The result is then known.The smaller class of uniformly supported operators is also closed under compo-sition.Lemma3.The composition P Q=(P x Q x,x∈G(0))of two uniformly supported families of operators P=(P x,x∈G(0))and Q=(Q x,x∈G(0))is uniformly supported.Proof.The reduced support suppµ(P Q)(see(12))of the composition P Q satisfiessuppµ(P Q)⊂µ suppµ(P)×suppµ(Q)whereµis the composition of arrows.Since suppµ(P)and suppµ(Q)are compact, the equation above completes the proof of the lemma.Let G be an almost differentiable groupoid.The space of order m,invariant,uni-formly supported pseudodifferential operators on G,acting on sections of the vector bundle E will be denoted byΨm(G;E).We denoteΨ∞(G;E)=∪m∈ZΨm(G;E) andΨ−∞(G;E)=∩m∈ZΨm(G;E).Thus an operator P∈Ψm(G;E)is actually a differentiable family P=(P x,x∈G(0))of ordinary pseudodifferential operators. Theorem1.The setΨ∞(G;E)of uniformly supported invariant pseudodifferential operators on an almost differentiable groupoid G is afiltered algebra,i.e.Ψm(G;E)Ψm′(G;E)⊂Ψm+m′(G;E).In particularΨ−∞(G;E)is a two-sided ideal.10V.NISTOR,A.WEINSTEIN,AND PING XUProof.Let P=(P x,x∈G(0))and Q=(Q x,x∈G(0))be two invariant uniformly supported pseudodifferential operators on G,of order m and m′respectively.Their composition P Q=(P x Q x),is a uniformly supported operator of order m+m′,in view of Lemma3.It is also a differentiable family due to Lemma2.We now check the invariance condition.Let g be an arbitrary arrow and U g:C∞c(G x,r∗(E))→C∞c(G y,r∗(E)),x=d(g)and y=r(g),be as in the definition above.Then(P Q)y U g=P y Q y U g=P y U g Q x=U g P x Q x=U g(P Q)x.This proves the theorem.Properly supported invariant differentiable families of pseudodifferential opera-tors also form afiltered algebra,denotedΨ∞prop(G;E).While it is clear that in order for our class of pseudodifferential operators to form an algebra we need some con-dition on the support of their distribution kernels,exactly what support condition to impose is a matter of choice.We prefer the uniform support condition because it leads to a better control at infinity of the family of operators P=(P x,x∈G(0)) and allows us to identify the regularizing ideal(i.e.the ideal of order−∞operators) with the groupoid convolution algebra of G.The choice of uniform support will also ensure thatΨm(G;E)behaves functorially with respect to open embeddings.The compact support condition enjoys the same properties but is usually too restrictive. The issue of support will be discussed again in examples.The definition of the principal symbol extends easily toΨm(G;E).Denote by π:A∗(G)→M,(M=G(0))the projection.If P=(P x,x∈G(0))∈Ψm(G;E)is an order m pseudodifferential differential operator on G,then the principal symbol σm(P)of P will be represented by sections of the bundle End(π∗E)and will be defined to satisfyσm(P)(ξ)=σm(P x)(ξ)∈End(E x)ifξ∈A∗x(G)=T∗x G x(13)(the equation above is mod S m−1cl (A∗x(G);End(E))).This equation will obviouslyuniquely determine a linear mapσm:Ψm(G)→S m cl(A∗(G);End(E))/S m−1cl(A∗(G);End(E)).provided we can show that for any P=(P x,x∈G(0))there exists a symbol a∈S m cl(A∗(G);End(E))whose restriction to A∗x(G)is a representative of the prin-cipal symbol of P x in thatfiber for each x.We thus need to choose for each P x a representative a x∈S m cl(A∗x(G);End(E))ofσm(P x)such that the family a x is smooth and invariant.Assumefirst that E is the trivial line bundle and proceed as in[14]Section18.1,especially Equation(18.1.27)and below.Choose a connection∇on the vector bundle A(G)→G(0)and consider the pull-back vector bundle r∗(A)→G of A(G)→G(0)endowed with the pull-back connection ∇=r∗∇.Its restriction on anyfiber G x defines a linear connection in the usual sense,which is denoted by∇x.These connections are right invariant in the sense thatR∗g∇x=∇y,∀g∈G such that r(g)=x and d(g)=y.(14)Using such an invariant connection,we may define the exponential map of a Lie algebroid,which generalizes the usual exponential map of a manifold with a connection and the exponential map of a Lie algebra as follows.For any x∈G(0),PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS11 define a map exp x:A x→G as the composition of the maps:A x i−→T x G x˜exp x−−−−→Gwhere i is the natural inclusion and˜exp x=exp∇x is the usual exponential mapat x∈G x on the manifold G x.By varying the point x,we obtain a map exp∇defined in a neighborhood of the zero section,called the exponential map of the Lie algebroid2.Clearly,exp∇is a local diffeomorphismA(G)⊃V0∋v−→exp∇(v)=y∈V⊂G(15)mapping an open neighborhood V0of the zero section in A(G)diffeomorphically to a neighborhood V of G(0)in G,and sending the zero section onto the set of units. Choose a cut-offfunctionφ∈C∞(G)with support in V and equal to1in a smaller neighborhood of G(0)in G.If y∈V,x=d(y)andξ∈A∗x(G)let v∈V0be the unique vector v∈A x(G)such that y=exp∇(v)and denote eξ(y)=φ(y)e iv·ξwhich extends then to all y∈G due to the cut-offfunctionφ.Define the(∇,φ)–complete symbolσ∇,φ(P)byσ∇,φ(P)(ξ)=(P x eξ)(x),∀ξ∈T∗x G x=A∗x(G).(16)Lemma4.If P=(P x,x∈G(0))is an operator inΨm(G)then the function σ∇,φ(P)defined above is differentiable and defines a symbol in S m cl(A∗(G)).More-over if(∇1,φ1)is another pair consisting of an invariant connection∇1and a cut-offfunctionφ1thenσ∇,φ(P)−σ∇,φ1(P)is in S−∞cl(A∗(G))andσ∇,φ(P)−σ∇1,φ1(P)is in S m−1cl(A∗(G)).Proof.For eachξ∈A∗x the function eξis smooth with compact support on G x so P x eξis defined.Equation(18.1.27)of[14]shows that a(ξ)=σ∇,φ(P)(ξ)is the restriction of the complete symbol of P xφto T∗x G x if the complete symbol is defined in the normal coordinate system at x∈G x(given by the exponential map). The normal coordinate system defines,using a local trivialization of A(G),afiber preserving diffeomorphismψ:d(V)×W→V for some open subset W of R n(i.e. satisfying d(ψ(x,w))=x).From the definition of the smoothness of the family P x (Definition6)it follows that the complete symbol of Pφis in S m cl(d(V)×T∗W)if the support ofφis chosen to be in V.This proves thatσ∇,φ(P)is in S m cl(A∗(G)).The rest follows in exactly the same way.The lemma above justifies the following definition of the principal symbol as the class ofσ∇,φ(P)modulo terms of lower order(for the trivial line bundle E=C). This definition will be,in view of the same lemma,independent on the choice of ∇orφand will satisfy Equation(13).If E is not trivial one can still define a complete symbolσ∇,∇′,φ(P),depending also on a second connection∇′on the bundle E,which is used to trivialize r∗(E)on V⊂G(assuming also that V0is convex).Alternatively,we can use Proposition3below.Proposition1.Let∇andφbe as above.The choice of a connection∇′on E defines a complete symbol mapσ∇,∇′,φ:Ψm(G;E)→S m cl(A∗(G)).The principalsymbolσm:Ψm(G;E)→S m cl(A∗(G))/S m−1cl(A∗(G)),defined byσm(P)=σ∇,∇′,φ(P)+S m−1cl (A∗(G))(17)。
