Introduction Groups and Organizations

An Introduction to Lie Groups and Symplectic Geometry Aseriesofnine lectures on Lie groups and symplectic geometry delivered at the Regional Geometry Institute in Park City, Utah, 24 June–20 July 1991. by Robert L. Bryant Duke University Durham, NC bryant@math.duke.edu This is an unofficial version of the notes and was last modified on 20 September 1993. The .dvi file for this preprint will be available by anonymous ftp from publications.math.duke.edu in the directory bryant until the manuscript is accepted for publication. You should get the ReadMe file first to see if the version there is more recent than this one. Please send any comments, corrections or bug reports to the above e-mail address. Introduction These are the lecture notes for a short course entitled “Introduction to Lie groups and symplectic geometry” which I gave at the 1991 Regional Geometry Institute at Park City, Utah starting on 24 June and ending on 11 July. The course really was designed to be an introduction, aimed at an audience of stu- dents who were familiar with basic constructions in differential topology and rudimentary differential geometry, who wanted to get a feel for Lie groups and symplectic geometry. My purpose was not to provide an exhaustive treatment of either Lie groups, which would have been impossible even if I had had an entire year, or of symplectic manifolds, which has lately undergone something of a revolution. Instead, I tried to provide an introduction to what I regard as the basic concepts of the two subjects, with an emphasis on examples which drove the development of the theory. Ideliberately tried to include a few topics which are not part of the mainstream subject, such as Lie’s reduction of order for differential equations and its relation with the notion of a solvable group on the one hand and integration of ODE by quadrature on the other. I also tried, in the later lectures to introduce the reader to some of the global methods which are now becoming so importantinsymplectic geometry. However, a full treatment of these topics in thespaceofnine lectures beginning at the elementary level wasbeyond my abilities. After the lectures were over, I contemplated reworking these notes into a comprehen- sive introduction to modern symplectic geometry and, after some soul-searching, finally decided against this. Thus, I have contented myself with making only minor modifications and corrections, with the hope that an interested person could read these notes in a few weeks and get some sense of what the subject was about. An essential feature of the course was the exercise sets. Each set begins with elemen- tary material and works up to more involved anddelicate problems. My object was to provide a path to understanding of the material which could be entered at several different levels and so the exercises vary greatly in difficulty. Many of these exercise sets are obvi- ously too long for any person to do them during the three weeks the course, so I provided extensive hints to aid the student in completing the exercises after the course was over. Iwanttotake this opportunity to thank the many people who made helpful sugges- tions for these notes both during and after the course. Particular thanks goes to Karen Uhlenbeck and Dan Freed, who invited me to give an introductory set of lectures at the RGI, and to my course assistant, Tom Ivey, who provided invaluable help and criticism in the early stages of the notes and tirelessly helped the students with the exercises. While the faults of the presentation are entirely myown,without the help, encouragement, and proofreading contributed by these Introduction Groups and Organizations Introduction Groups and Organizations Bởi: OpenStaxCollege The alternative punk band NOFX is playing outside in Los Angeles The music is loud, the crowd pumped up and excited But neither the lyrics nor the people in the audience are quite what you might expect Mixed in with the punks and young rebel students are members of local unions, from well-dressed teachers to more grizzled labor leaders The lyrics are not published anywhere but are available on YouTube: “We’re here to represent/The 99 percent/Occupy, occupy, occupy.” The song: “Wouldn’t It Be Nice If Every Movement Had a Theme Song” (Cabrel 2011) Across the country at an Occupy camp in New York, roughly three dozen members of the Facilitation Working Group, a part of the General Assembly, take a steady stream of visitors with requests at their unofficial headquarters One person wants a grant for $1500 to make herbal medications available to those staying at the park Another wants to present Native American peace principles derived from the Iroquois Confederacy Yet another has a spreadsheet that he wants used as an evaluation tool for the facilitators Numerous groups make up this movement, yet there’s no national leader What makes a group something more than just a collection of people? How are leadership functions and styles established in a group dynamic? 1/3 Introduction Groups and Organizations Nurses, teachers, and Teamsters showed up to protest at the Occupy movement (Photo courtesy of David Shankbone/flickr) Most people have a sense of what it means to be a part of some kind of a group, whether it is a sports team, sorority, school club, or family Groups connect us to others through commonalities of geography, interests, race, religion, and activities But for the groups of people protesting from Augusta, Georgia, to Oakland, California, and hundreds of cities in between, their connection within the Occupy Wall Street movement is harder to define What unites these people? Are the out-of-work doctoral candidates truly aligned with the high school dropouts? Do the urban poor genuinely feel for the campus-based protest against university tuition hikes? Groups are prevalent in our social lives and provide a significant way we understand and define ourselves—both through groups we feel a connection to and those we don’t Groups also play an important role in society As enduring social units, they help foster shared value systems and are key to the structure of society as we know it There are three primary sociological perspectives for studying groups: Functionalist, Conflict, and Interactionist We can look at the Occupy movement through the lenses of these methods to better understand the roles and challenges that groups offer The Functionalist perspective is a big-picture macro-level view that looks at how different aspects of society are intertwined This perspective is based on the idea that society is a well-balanced system with all parts necessary to the whole, and it studies the roles these parts play in relation to the whole In the case of the Occupy Movement, a Functionalist might look at what macro-level needs the movement serves For example, a Structural Functionalist might ask how the Occupy Wall Street movement forces both haves and have-nots to pay attention to the economy, or the way urbanites are impacted by the influx of protestors who typically reside outside of their region The Conflict perspective is another macroanalytical view, one that focuses on the genesis and growth of inequality A conflict theorist studying the Occupy movement might look at how business interests have manipulated the system over the last 30 years, leading to the gross inequality we see today Or this perspective might explore how the massive redistribution of wealth from the middle class to the upper class could lead to a two-class system reminiscent of Marxist ideas A third perspective is the Symbolic Interaction or Interactionist perspective This method of analyzing groups takes a micro-level view Instead of studying the big picture, these researchers look at the day-to-day interactions of groups Studying these details, the Interactionist looks at issues like leadership style and group dynamics In the case of the Occupy Movement, Interactionists might ask, “How does the group dynamic in New York differ from that in Atlanta?” Or, “What dictates who becomes the de facto leader in different cities—geography, social dynamics, economic circumstances?” 2/3 Introduction Groups and Organizations References Cabrel, Javier 2011 “NOFX - Occupy LA.” LAWeekly.com, November 28 Retrieved February 10, 2012 ([link]) 3/3 arXiv:math-ph/0005032 31 May 2000 An Elementary Introduction to Groups and Representations Brian C. Hall Author address: University of Notre Dame, Department of Mathematics, Notre Dame IN 46556 USA E-mail address: bhall@nd.edu Contents 1. Preface ii Chapter 1. Groups 1 1. Definition of a Group, and Basic Properties 1 2. Some Examples of Groups 3 3. Subgroups, the Center, and Direct Products 4 4. Homomorphisms and Isomorphisms 5 5. Exercises 6 Chapter 2. Matrix Lie Groups 9 1. Definition of a Matrix Lie Group 9 2. Examples of Matrix Lie Groups 10 3. Compactness 15 4. Connectedness 16 5. Simple-connectedness 18 6. Homomorphisms and Isomorphisms 19 7. Lie Groups 20 8. Exercises 22 Chapter 3. Lie Algebras and the Exponential Mapping 27 1. The Matrix Exponential 27 2. Computing the Exponential of a Matrix 29 3. The Matrix Logarithm 31 4. Further Properties of the Matrix Exponential 34 5. The Lie Algebra of a Matrix Lie Group 36 6. Properties of the Lie Algebra 40 7. The Exponential Mapping 44 8. Lie Algebras 46 9. The Complexification of a Real Lie Algebra 48 10. Exercises 50 Chapter 4. The Baker-Campbell-Hausdorff Formula 53 1. The Baker-Campbell-Hausdorff Formula for the Heisenberg Group 53 2. The General Baker-Campbell-Hausdorff Formula 56 3. The Series Form of the Baker-Campbell-Hausdorff Formula 63 4. Subgroups and Subalgebras 64 5. Exercises 65 Chapter 5. Basic Representation Theory 67 1. Representations 67 2. Why Study Representations? 69 iii iv CONTENTS 3. Examples of Representations 70 4. The Irreducible Representations of su(2) 75 5. Direct Sums of Representations and Complete Reducibility 79 6. Tensor Products of Representations 82 7. Schur’s Lemma 86 8. Group Versus Lie Algebra Representations 88 9. Covering Groups 94 10. Exercises 96 Chapter 6. The Representations of SU(3), and Beyond 101 1. Preliminaries 101 2. Weights and Roots 103 3. Highest Weights and the Classification Theorem 105 4. Proof of the Classification Theorem 107 5. An Example: Highest Weight (1, 1) 111 6. The Weyl Group 112 7. Complex Semisimple Lie Algebras 115 8. Exercises 117 Chapter 7. Cumulative exercises 119 Chapter 8. Bibliography 121 1. PREFACE v 1. Preface These notes are the outgrowth of a graduate course on Lie groups I taught at the University of Virginia in 1994. In trying to find a text for the course I discovered that books on Lie groups either presuppose a knowledge of differentiable manifolds or provide a mini-course on them at the beginning. Since my students did not have the necessary background on manifolds, I faced a dilemma: either use manifold techniques that my students were not familiar with, or else spend much of the course teaching those techniques instead of teaching Lie theory. To resolve this dilemma I chose to write my own notes using the notion of a matrix Lie group. A matrix Lie group is simply a closed subgroup of GL(n; C). Although these are often called simply “matrix groups,” my terminology emphasizes that every matrix group is a Lie group. This approach to the subject allows me to get started quickly on Lie group the- ory proper, with a minimum of prerequisites. Since most of the interesting examples of Lie groups are matrix Lie groups, there is not too much loss of generality. Fur- thermore, the proofs of the main results are ultimately similar to standard proofs in the general setting, but with less preparation. Of course, there is a price to be paid and certain constructions (e.g. covering groups) that are easy in the Lie group setting are problematic in the matrix group setting. (Indeed the universal cover of a matrix Lie group need not be a matrix Lie group.) On the other hand, the matrix approach suffices for a first course. Anyone planning to do ninth edition STEPHEN P. ROBBINS © 2007 Prentice Hall, Inc. © 2007 Prentice Hall, Inc. All rights reserved. All rights reserved. PowerPoint Presentation by Charlie Cook PowerPoint Presentation by Charlie Cook The University of West Alabama The University of West Alabama MARY COULTER Introduction to Introduction to Management and Management and Organizations Organizations Chapter Chapter 1 1 © 2007 Prentice Hall, Inc. All rights reserved. 1–2 L E A R N I N G O U T L I N E L E A R N I N G O U T L I N E Follow this Learning Outline as you read and study this chapter. Follow this Learning Outline as you read and study this chapter. Who Are Managers? Who Are Managers? • Explain how managers differ from non-managerial Explain how managers differ from non-managerial employees. employees. • Describe how to classify managers in organizations. Describe how to classify managers in organizations. What Is Management? What Is Management? • Define management. Define management. • Explain why efficiency and effectiveness are important to Explain why efficiency and effectiveness are important to management. management. © 2007 Prentice Hall, Inc. All rights reserved. 1–3 L E A R N I N G O U T L I N E (cont’d) L E A R N I N G O U T L I N E (cont’d) Follow this Learning Outline as you read and study this chapter. Follow this Learning Outline as you read and study this chapter. What Do Managers Do? What Do Managers Do? • Describe the four functions of management. Describe the four functions of management. • Explain Mintzberg’s managerial roles. Explain Mintzberg’s managerial roles. • Describe Katz’s three essential managerial skills and how Describe Katz’s three essential managerial skills and how the importance of these skills changes depending on the importance of these skills changes depending on managerial level. managerial level. • Discuss the changes that are impacting managers’ jobs. Discuss the changes that are impacting managers’ jobs. • Explain why customer service and innovation are Explain why customer service and innovation are important to the manager’s job. important to the manager’s job. © 2007 Prentice Hall, Inc. All rights reserved. 1–4 L E A R N I N G O U T L I N E (cont’d) L E A R N I N G O U T L I N E (cont’d) Follow this Learning Outline as you read and study this chapter. Follow this Learning Outline as you read and study this chapter. What Is An Organization? What Is An Organization? • Describe the characteristics of an organization. Describe the characteristics of an organization. • Explain how the concept of an organization is changing. Explain how the concept of an organization is changing. Why Study Management? Why Study Management? • Explain the universality of management concept. Explain the universality of management concept. • Discuss why an understanding of management is Discuss why an understanding of management is important. important. • Describe the rewards and challenges of being a manager. Describe the rewards and challenges of being a manager. © 2007 Prentice Hall, Inc. All rights reserved. 1–5 Who Are Managers? Who Are Managers? • Manager Manager  Someone who coordinates and oversees the work of Someone who coordinates and oversees the work of other people so that organizational goals can be other people so that organizational goals can be accomplished. accomplished. © 2007 Prentice Hall, Inc. All rights reserved. 1–6 Classifying Managers Classifying Managers • First-line Managers First-line Managers  Individuals who manage the work of non-managerial Individuals who manage the work of non-managerial employees. employees. • Middle Managers Middle Managers  Individuals who manage the work of first-line Individuals who manage the work of first-line managers. ninth edition STEPHEN P ROBBINS Chapter © 2007 Prentice Hall, Inc All rights reserved MARY COULTER Introduction to Management and Organizations PowerPoint Presentation by Charlie Cook The University of West Alabama LEARNING OUTLINE Follow this Learning Outline as you read and study this chapter Who Are Managers? • Explain how managers differ from non-managerial employees • Describe how to classify managers in organizations What Is Management? • Define management • Explain why efficiency and effectiveness are important to management © 2007 Prentice Hall, Inc All rights reserved 1–2 L E A R N I N G O U T L I N E (cont’d) Follow this Learning Outline as you read and study this chapter What Do Managers Do? • Describe the four functions of management • Explain Mintzberg’s managerial roles • Describe Katz’s three essential managerial skills and how the importance of these skills changes depending on managerial level • Discuss the changes that are impacting managers’ jobs • Explain why customer service and innovation are important to the manager’s job © 2007 Prentice Hall, Inc All rights reserved 1–3 L E A R N I N G O U T L I N E (cont’d) Follow this Learning Outline as you read and study this chapter What Is An Organization? • Describe the characteristics of an organization • Explain how the concept of an organization is changing Why Study Management? • Explain the universality of management concept • Discuss why an understanding of management is important • Describe the rewards and challenges of being a manager © 2007 Prentice Hall, Inc All rights reserved 1–4 Who Are Managers? • Manager  Someone who coordinates and oversees the work of other people so that organizational goals can be accomplished © 2007 Prentice Hall, Inc All rights reserved 1–5 Classifying Managers • First-line Managers  Individuals who manage the work of non-managerial employees • Middle Managers  Individuals who manage the work of first-line managers • Top Managers  Individuals who are responsible for making organization-wide decisions and establishing plans and goals that affect the entire organization © 2007 Prentice Hall, Inc All rights reserved 1–6 Exhibit 1–1 Managerial Levels © 2007 Prentice Hall, Inc All rights reserved 1–7 What Is Management? • Managerial Concerns  Efficiency  “Doing things right” – Getting the most output for the least inputs  Effectiveness  “Doing the right things” – Attaining organizational goals © 2007 Prentice Hall, Inc All rights reserved 1–8 Exhibit 1–2 Effectiveness and Efficiency in Management © 2007 Prentice Hall, Inc All rights reserved 1–9 What Do Managers Do? • Functional Approach  Planning  Defining goals, establishing strategies to achieve goals, developing plans to integrate and coordinate activities  Organizing  Arranging and structuring work to accomplish organizational goals  Leading  Working with and through people to accomplish goals  Controlling  Monitoring, comparing, and correcting work © 2007 Prentice Hall, Inc All rights reserved 1–10 Exhibit 1–6 Conceptual Skills • Using information to solve business problems • Identifying of opportunities for innovation • Recognizing problem areas and implementing solutions • Selecting critical information from masses of data • Understanding of business uses of technology • Understanding of organization’s business model Source: Based on American Management Association Survey of Managerial Skills and Competencies, March/April 2000, found on AMA Web site (www.ama.org), October 30, 2002 © 2007 Prentice Hall, Inc All rights reserved 1–16 Exhibit 1–6 Communication Skills • Ability to transform ideas into words and actions • Credibility among colleagues, peers, and subordinates • Listening and asking questions • Presentation skills; spoken format • Presentation skills; written and/or graphic formats Source: Based on American Management Association Survey of Managerial Skills and Competencies, March/April 2000, found on AMA Web site (www.ama.org), October 30, 2002 © 126 Test Bank for Organizational Behavior Integrating Individuals Groups and Organizations 3rd Edition Champoux Multiple Choice Questions - Page _ coined the phrase gold in the mine to refer to long-term payoffs of quality management A Walter A Shewhart B Armand V Feigenbaum C Joseph M Juran D W Edwards Deming The expected gender profile of the labor force in 2010 shows A 25% women B 48% women C 64% women D 75% women Quality management’s roots lie in A service B manufacturing C customer satisfaction D the public sector Quality management’s roots can be traced to the A 1920s B 1940s C 1970s D 1980s Which of the following is true about the technology context of modern organizations? A Managers now have more organizational design options to build flexible, responsive organizations than ever before B Computing technologies have reshaped manufacturing processes worldwide C Communication technologies let people stay connected, no matter where they are in the world D All answers are true about the technology context of modern organizations Which of the following is not a quality management tool or technique? A flowchart B path analysis C benchmarking D Pareto chart Which of the following is not true about the context of modern organizations? A technology effects on organizations B decreasing global emphasis of managers C increasing workforce diversity D emphasis on managing for quality Quality management applies to which of the following aspects of organizations? A production B service C processes D Each answer is an organizational aspect to which quality management applies Which of the following is true about the workforce diversity context of modern organizations? A Workforce diversity brings people into organizations with vastly different ways of thinking about their environment B Managers need to harness those differences to reach organizational goals successfully C Workforce diversity can increase an organization’s conflict potential D All answers are true about the workforce diversity context of modern organizations Genichi Taguchi is a major Japanese contributor to quality management Which of the followingperspectives or tools did Taguchi bring to QM? A He viewed any lack of quality as a loss to society B Taguchi developed a “loss function” to measure the costs of quality losses that decision makers could use to guide their QM efforts C Taguchi applied experimental design to quality assessment D Each answer is a perspective or tool from Taguchi The person who developed a quality control method that involved people both within and outside a quality control department was A Walter A Shewhart B Armand V Feigenbaum C Joseph M Juran D W Edwards Deming Which of the following is not a quality management tool or technique? A control chart B checksheet C variance analysis D cause and effect diagram _ was the first to recognize that process abnormalities caused variability in manufacturing processes A Walter A Shewhart B Armand V Feigenbaum C Joseph M Juran D W Edwards Deming Which of the of the following people the Japanese view as the most important contributor to their quality management methods? A Walter A Shewhart B Armand V Feigenbaum C Joseph M Juran D W Edwards Deming American organizations started to embrace quality management in the A 1920s B 1940s C 1970s D 1980s Which of the following statements is true about quality management? A A goal of quality management is increased profit B A goal of quality management is long-term cost reduction C A goal of quality management is statistical control D A goal of quality management is continuous quality improvement Which of the following statements is not true about managing ... tuition hikes? Groups are prevalent in our social lives and provide a significant way we understand and define ourselves—both through groups we feel a connection to and those we don’t Groups also.. .Introduction Groups and Organizations Nurses, teachers, and Teamsters showed up to protest at the Occupy movement (Photo courtesy... family Groups connect us to others through commonalities of geography, interests, race, religion, and activities But for the groups of people protesting from Augusta, Georgia, to Oakland, California,