Course Code : | 2600WETCAF |

Study domain: | Mathematics |

Academic year: | 2019-2020 |

Semester: | 1st semester |

Contact hours: | 60 |

Credits: | 6 |

Study load (hours): | 168 |

Contract restrictions: | No contract restriction |

Language of instruction: | Dutch |

Exam period: | exam in the 1st semester |

Lecturer(s) | David Eelbode |

At the start of this course the student should have acquired the following competences:

specific prerequisites for this course

specific prerequisites for this course

Having a credit for a course in Complex Analysis (Bachelor)

- Knowing how to classify the real Clifford algebras (technical expertise)
- Being able to construct spinors and to explain what their role is in theoretical physics (technical expertise)
- Being able to explain what the classical Howe duality has to say about the orthogonal group (technical expertise)
- Being able to illustrate in what sense Clifford analysis defines a generalisation of classical complex analysis (technical expertise)
- Knowing how to apply algebraic techniques in the setting of elementary particle physics (interdisciplinary knowledge)
- Being able to illustrate the general theory of root systems for Lie algebras in terms of the orthogonal Lie algebra and some of her representations (interdisciplinary knowledge)
- Being able to explain connections between analysis in higher dimensions and Lie theory (interdisciplinary knowledge)

This course is devoted to topics lying on the crossroad of multivariable analysis, representation theory for Lie algebras and theoretical physics. Starting from 2 differential equations playing a crucial role in physics (the Klein-Gordon and Dirac equation for massless particles), we will introduce notions which are useful in both abstract algebra and theoretical physics. For each of these equations, we will study the symmetries, the vector spaces of homogeneous polynomial solutions and their role as representation spaces for the orthogonal Lie algebra, the dual Howe symmetry behind these equations and a few formulas generalising what one already knows from complex analysis (e.g. Cauchy's formula).

In order to do this, we will have to define the Clifford algebra's and their representations: this will allow the introduction of spinors (important when studying the Dirac equation), and simple models for the root system of the orthogonal Lie algebra.

Class contact teachingLectures

Directed self-study

**5.3 Facilities for working students ***

Classroom activities

Directed self-study (possibly with response lecture)

Directed self-study

Classroom activities

- Seminars/tutorials: alternative assignment possible

Directed self-study (possibly with response lecture)

- Blended learning with limited amount of classroom activities in the evening

ExaminationOral with written preparation Closed book Open-question

Course notes

David Eelbode

(david.eelbode@uantwerpen.be)