2 semester

1

Course Title

Introduction to Lie group theory  

2

Year of study, speciality 

1, speciality  “Mathematics”

3

Semester of study

2

4

Credit points

4

5

Name of the lecturer, scientific degree, occupation

Associate Professor Vitaly V. Balashchenko,

PhD in Mathematics, Department of Geometry, Topology and Methods of Teaching Mathematics

6

Goals of studying

Studying one of the most important classes of mathematical objects – smooth manifolds, which are widely used in theoretical and applied directions of mathematics. Based on this information, the main goal of the subsequent presentation is the introduction to Lie group theory, which is one of the most advanced branches of modern mathematics and is widely used not only within mathematics, but also in mechanics, theoretical physics and other natural sciences.

7

Prerequisites

Differential geometry and topology,  algebra, mathematical analysis, ordinary and partial differential equations

8

Contents

Smooth manifolds, examples. Smooth functions and smooth mappings, diffeomorphism. Tangent vectors, the tangent space. Smooth vector fields, the Lie algebra of smooth vector fields. Fibre bundles, bundle morphisms. The tangent bundle. Lie groups, examples. Direct product of Lie groups, homomorphisms. The Lie algebra of a Lie group. Linear Lie groups. Classical Lie groups and classical geometries.

9

Recommended literature

  1. Warner F. Foundations of Differentiable Manifolds and Lie Groups. – Springer, 1983. 
  2. Vinberg E.B. Linear Representations of Groups. – Birkhäuser, Boston, 1989.
  3. Helgason S. Differential Geometry, Lie Groups, and Symmetric Spaces. – Academic Press, London, 1978.

10

Teaching methodology

Lectures, solving problems, discussions, consultations

11

Language of instruction  

Russian

12

Requirements for study during the semester

  • individual and collective tasks;
  • short tests.

The final evaluation is made taking into account:

  • 30% – assessment of current academic performance;
  • 70% – result during the exam

13

Examination methodology

Oral examination