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Web page of the mastermath course
Lie Groups and Lie Algebras

(Spring 2019)


Teachers: Eric M. Opdam and Jasper V. Stokman
Emails: e.m.opdam (AT) uva.nl and j.v.stokman (AT) uva.nl
Tel.: 020-5255205 and 020-5255202
Room numbers: F3.04 and F3.04 (Science Park 105-107, Amsterdam).

Course description

This mastermath course (8 EC) provides a basic introduction to the theory of Lie groups and Lie algebras.

Lie groups are differentiable manifolds with a compatible group structure. This notion was introduced in the late nineteenth century by Sophus Lie to capture and analyze continuous symmetries of differential equations. The group structure of the Lie group induces an algebraic structure on the tangent space at its unit element, turning it in a so-called Lie algebra. Lie algebras provide a powerful algebraic tool in the study of Lie groups and their algebraic geometric counterparts, the so-called algebraic groups.

In the course we introduce real Lie groups and their associated Lie algebras. We give basic examples and discuss Lie's fundamental theorems on the correspondences between Lie groups and Lie algebras. We treat the Killing-Cartan classification of semisimple Lie algebras. We give a first introduction to the representation theory of compact Lie groups and semisimple Lie algebras.

Schedule

Fridays, week 6-15 and 17-21 (start date: February 8, 2019 and end date: May 24, 2019).
Time: 10:00-12:45.
Location: Science Park Amsterdam, room C1.112 (Science Park 907, Amsterdam directions).

Literature

We follow the book

- A. Kirillov, Jr., "An Introduction to Lie groups and Lie algebras", Cambridge studies in advanced mathematics 113

(here is the online version). Occasionally we will use

- Erik van den Ban, Lecture notes "Lie groups"

and

-James E. Humphreys, "Introduction to Lie Algebras and Representation Theory", Graduate Texts in Mathematics 9, Springer Verlag

for further details of proofs and for exercises.

Prerequisites

1. Group and ring theory as covered by the standard algebra courses in the bachelor Mathematics.
2. Basic notions from differential geometry, as covered by a bachelor mathematics course on Differential Geometry (in particular, familiarity with notions like: differentiable manifolds, tangent space, vector fields, integral curves, immersions and submersions). These basic notions are shortly discussed in Erik van den Ban's Prerequisites from differential geometry.

Examination

A midterm and final written exam. The midterm exam counts for 30% of the final grade. In case of a re-exam, the midterm exam will be discarded.

The midterm exam will take place in week 15 (Friday, April 12), time 10:00-12:00, in room C1.112 (Science Park, Amsterdam).

The final exam will take place in week 25 (Friday, June 21), time 10:00-13:00, in room H0.08 (Science Park, Amsterdam).

The reexam will take place in week 28 (Friday, July 12), time 10:00-13:00, in room D1.112 (Science Park, Amsterdam).


Program

Here we list the program per week, including homework and recommended exercises.
As preparation for the first lecture, please refresh your knowledge on basic notions from differential geometry by carefully reading the Prerequisites from Differential Geometry by Erik van den Ban.

Week 6 (February 8): to be announced.