Lie groups (Mastermath, WISL 519, 2019)

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Announcements:
  • Office hours: January 16, 14:00- 17:00, HFG 604 (my office). To ask questions in preparation for the exam. Or to have a look at the homework exercises. Please send me an email if you intend to come.
  • Homework rule: out of the 8 HW scores, only the 7 best will be taken into account. See precise description below.
  • Change of starting time: on November 14, the lecture will be from 13:30 - 15:15. There will be no exercise class.
  • Change of lecture room: from November 14 on, we change back to Buys Ballot Building BG.023
  • Change of lecture room: on October 24, 31 and November 7, the lecture and exercise class will be given in Koningsberger KBG PANGEA.

    • Location and time:

    Utrecht University, Buys Ballot Building BG.023, Thursday 14:00 - 16:45, Weeks 37 - 51.
    Lecturer:
  • E.P. van den Ban (UU)

    • Material discussed in lectures

    Exercises



    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 expanded enormously in the course of the previous century. Nowadays, 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 tangent space at the identity, with a suitable bracket structure. 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 (2012 version)

    • Prerequisites:

  • Basic knowledge of analysis, topology and group theory.
  • Basic knowledge from the theory of differentiable manifolds: notion of smooth manifold, tangent spaces, vector field as in
    John M. Lee, Introduction to Smooth Manifolds:
  • Chapter 1, Smooth Manifolds (pp 1-24);
  • Chapter 2, Smooth Maps (pp 31-37, 49 - 55);
  • Chapter 3, Tangent vectors (pp. 60-73,75 - 78);
  • Chapter 4, Vector fields (pp. 81-92).
  • See also Prerequisites from differential geometry, for Lie groups.
  • In case of knowledge deficits, detailed reading instructions will be given by the lecturer.

    • 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.
  • John M. Lee
    Introduction to Smooth Manifolds
    Second edition. Graduate Texts in Mathematics, 218. Springer, New York, 2013. xvi+708 pp.
    ISBN: 978-1-4419-9981-8
  • Th. Frankel
    The Geometry of Physics-an introduction.
    Cambridge University Press, 1997.
  • E.J.N. Looijenga
    Smooth manifolds
    Lecture Notes, available as pdf file.

    • Exam:
    The exam consists of two parts:

  • Homework exercises.
  • Written exam at the end of the course.

    • Written exam:
    January 30, 2020, 13:30 - 16:30. Educatorium Megaron

  • This will be an open book exam. You are allowed to bring lecture notes and personal notes, but no exercises, worked exercises or notes on the exercises.
    • The final grade will be based on the grades obtained for the homework (40 percent) and the concluding exam (60 percent). Out of the 8 HW scores, only the 7 best will count for the homework average H (calculated in one decimal accuracy). The grade E for the concluding exam will be calculated with one decimal of accuracy. The final grade G is calculated by G = 0.4 H + 0.6 E, rounded off to an integer in the usual way.

    Retake:

    The re-exam may be oral, depending on the number of participants. In the case of an oral a date will be planned in mutual agreement.

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

    Last update: 11/7 - 2019