Group Theory

6 EC

Semester 2, period 4, 5

5354GRTH6Y

Owner Master Physics and Astronomy (joint degree)
Coordinator dr. W.J. Waalewijn
Part of Master Physics and Astronomy, track Theoretical Physics, year 1

Course manual 2018/2019

Course content

The course starts by introducing the basic notion of a group, its generators and relations. Important concepts such as cosets, conjugacy classes, and  (normal) subgroups will be explained. After briefly describing examples like the cyclic, dihedral, and Platonic groups, a more detailed discussion will be given of the permutation group S_N of N elements. Continuous groups are first introduced with the example of the two and three-dimensional rotation groups SO(2) and SO(3), but quickly extended to the classical groups, with a extensive discussion of SU(N). The theory of Lie algebras is explained, including a discussion of the Cartan-Weyl basis and the root lattice. The Cartan classification and the existence of exceptional groups is briefly discussed. A significant part of the course covers the representations of groups through matrices, which includes the use of the characters for finite groups, Young tableaux for SU(N) and the (general) construction of the unitary highest weight representations. The Lorentz group will also be discussed.

Objectives

  • Understand, apply and evaluate basic concepts: group axioms, cyclic, dihedral and permutation group, platonic solids, conjugacy, subgroups, cosets, homomorphism
  • Understand and apply representations of finite groups: irreducibility and equivalence, Schur’s lemmas, orthogonality of characters, character tables, product representation
  • Apply and analyze SO(N): generators and Lie algebra, characters and group measure, Clebsch-Gordon decomposition, Wigner-Eckart theorem
  • Know, understand and apply SU(N): isospin, invariant Levi-Civita tensor, irreducible representations from tensors and Young tableaux
  • Understand and apply Lie algebras: Cartan-Killing metric, adjoint representation, Cartan basis, roots, Cartan matrix, Dynkin diagrams, classification, weights

Teaching methods

  • Lecture
  • Self-study
  • Exercise session

Learning activities

Activity

Hours

Lectures

30

Exams

5

Exercise sessions

30

Self study

103

Total

168

(6 EC x 28 hours)

Attendance

Requirements concerning attendance (OER-B).

  • In addition to, or instead of, classes in the form of lectures, the elements of the master’s examination programme often include a practical component as defined in article 1.2 of part A. The course catalogue contains information on the types of classes in each part of the programme. Attendance during practical components is mandatory.

    • Assessment

    Item and weight Details

    Final grade

    0.4 (40%)

    Midterm

    0.6 (60%)

    Final

    You are not allowed to bring any literature, notes, calculators etc. to any of the exams. The retake is for the whole course.

    Inspection of assessed work

    After the grades of the exams are posted on canvas, we will schedule a time for students to inspect their work. The exam questions and solutions will be posted on canvas.

    Fraud and plagiarism

    The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

    Course structure

    WeeknummerOnderwerpenStudiestof
    1
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16

    Timetable

    The schedule for this course is published on DataNose.

    Contact information

    Coordinator

    • dr. W.J. Waalewijn