Lie Groups (Mastermath, WISM 414, 2012)

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Announcements:
  • Here is the anonymous list of grades for the course.
  • Here is the full anonymous list of homework grades.
  • For a copy of the exam, click here, and for the solutions, click here.
  • Final Lecture:Wednesday, June 30, 11:15 - 13:00. Lecture room 208, Minnaert.
  • Rules for the written exam: see bottom of page.
  • From May 9 on: no hand in exercises anymore. However all exercises form important training for the written exam.
  • Please check your homework grades in this file. If you cannot find your student number in this file, please send your name and student number to Liegroups2012[*at*]gmail.com
  • Exercises 23 and 24 contained misprints. These have now been corrected.
  • Please be aware of the following rescheduling of lectures and exercise class: exercise class 10:15 - 11:00, lectures 11:15 - 13:00. See below for further details.
  • From now in homework exercises may be handed in at the start of class two weeks after the date of announcement.
  • Please do not submit solutions of the homework exercises to the email addresses of the assistants, but use the following email address instead:
    Liegroups2012[*at*]gmail.com
  • The web movies of the lectures are available.
  • Change of location To accomodate the larger expected number of attending students, the lecture room has been changed to room 169 of the Buys Ballot Building.
  • The course starts later than announced originally:
    First lecture: Wednesday, Feb 15 , 10:15 - 13:00.
  • See Material discussed in lectures for the content of the first lecture.
  • Homework exercises are indicated with a star, and should be handed in at the start of class two weeks later.
  • Homework prepared in (La)TeX and in pdf format may be sent by email to Liegroups2012[*at*]gmail.com

  • Location and time:

  • Exercise class: Wednesdays 10:15 - 11:00. Location: University of Utrecht, Buys Ballot Building (BBL) room 169.
  • Lectures: Wednesdays 11:15 - 13:00. Location: UU, alternating lecture rooms:
  • weeks 10, 11, 15, 18 t/m 21: Aardwetenschappen Klein
  • weeks 12, 17: Aardwetenschappen Groot
  • weeks 13, 14: BBL 169
  • weeks 16, 22: Minnaert building, 208
  • Period 3 and 4: weeks 7 - 22.
  • First lecture: February 15, 2012.
  • Last lecture: May 30, 2012.
  • Exam: Homework exercises and written exam, see bottom of the page for details.
  • Teacher:
  • E.P. van den Ban (UU)
  • Instructors:
  • Joao Nuno Mestre, jnmestre(symbol: at)gmail.com
  • Boris Osorno Torres, bosornot(symnbol: at)gmail.com
  • Ori Yudilevich, oriyudilevich(symbol: at)gmail.com
  • E.P. van den Ban (UU)

    Material discussed in lectures
    Exercises
    Web movies of lectures

    Aim:
    The aim of this course is to give a thorough introduction to the theory of Lie groups and algebras.

    Course description:
    A Lie group is a group with the additional structure of a differentiable manifold for which the group operation is differentiable. The name Lie group comes from the Norwegian mathematician M. Sophus Lie (1842-1899) who was the first to study these groups systematically in the context of symmetries of partial differential equations.
    The theory of Lie groups has developed vastly in the course of the previous century. It plays a vital role in the description of symmetries in Physics (quantum physics, elementary particles), Geometry and Topology, and Number Theory (automorphic forms).
    In this course we will begin by studying the basic properties of Lie groups. Much of the structure of a connected Lie group is captured by its Lie algebra, which may be defined as the algebra of left invariant vector fields. The exponential map will be introduced, and the relation between the structure of a Lie group and its Lie algebra will be investigated. Actions of Lie groups will be studied.
    After this introduction we will focus on compact Lie groups and the integration theory on them. The groups SU(2) and SO(3) will be discussed as basic examples. We will study representation theory and its role in the harmonic analysis on a Lie group. The classifiction of the irreducible representations of SU(2) will be studied.
    The final part of the course will be devoted to the classification of compact Lie algebras. Key words are: root system, finite reflection group, Cartan matrix, Dynkin diagram. The course will be concluded with the formulation of the classification of irreducible representations by their highest weight and with the formulation of Weyl's character formula.

    Text:

  • Lecture Notes (2010 version)
  • Exercise collection (2009 version)
  • Prerequisites:

  • Basic knowledge of analysis, topology and group theory, as taught in the Bachelor programme.
  • Basic knowledge from the theory of differentiable manifolds: smooth manifold, tangent spaces, vector fields and flow. Immersion and submersion theorem. In Utrecht, these subjects are taught in the Bachelor course WISB 342: Differentieerbare varieteiten.
  • See also Prerequisites from differential geometry, for Lie groups.
  • Recommended literature:

  • T. Br"ocker & T. tom Dieck
    Representations of compact Lie groups,
    Springer-Verlag, New York, 1985.
  • W. Rossmann
    Lie groups: An Introduction Through Linear Groups
    Oxford Graduate Texts in Mathematics, Number 5.
    Oxford University Press, 2002; ISBN 0198596839
  • J.J. Duistermaat & J.A.C. Kolk
    Lie groups
    Universitext serie, Springer-Verlag, New York, 2000.
    ISBN 3-540-15293-8, cat prijs DM 79.
  • Th. Frankel
    The Geometry of Physics-an introduction.
    Cambridge University Press, 1997.
  • E.J.N. Looijenga
    Meetkunde op varieteiten
    diktaat, available as ps file of als pdf file.
  • Exam: The exam consists of two parts:

  • Homework exercises.
  • Written exam at the end of the course:
  • The exam covers theory and exercises of weeks 7 - 21 (not week 22).
  • It is an open book exam. You are allowed to bring the lectured notes, but no worked exercises.
  • Date and time exam: June 6, 13:30-16:30, Educatorium Alfa-zaal.
  • The final grade will be based on the grades obtained for the homework (40 percent) and the concluding exam (60 percent).


    E.P.vandenBan[*at*]uu.nl

    Last update: April 27, 2012