Course manual 2022/2023

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 and its representations will also be discussed.

Study materials

Literature

Group Theory in a Nutshell for Physicists, A. Zee

Objectives

Understand, apply and evaluate basic concepts: group axioms, cyclic, dihedral and permutation group, 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, Lorentz group and its representations
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 and Dynkin diagrams, classification, representations and weights

Teaching methods

Lecture
Self-study
Exercise session

In the lecture the main ideas of the course, as well as some examples, are presented. Further details are worked out during self-study and the exercise sessions, where these ideas are put to practice.

Learning activities

Activity	Hours
Lectures	30
Exams	3
Exercise sessions	28
Self study	107
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 A-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
0.8 (80%) Tentamen
0.2 (20%) Homework

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.

Assignments

There will be an assessed assignment every other week that students should make individually.

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

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

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

Group Theory