Course Identification

Title:

Lie groups

Code:

20264122

Lecturers and Teaching Assistants

Lecturers:

Prof. Vladimir Hinich

TA's:

N/A

Course Schedule and Location

Year:

2026

Semester:

Second Semester

When / Where:

Tuesday, 11:00 - 13:00, Goldsmith, room 108
Tuesday, 14:00 - 15:00, Jacob Ziskind Building, Rm 155

First Lecture:

17/03/2026

End date:

23/06/2026

Field of Study, Course Type and Credit Points

Mathematics and Computer Science: Lecture; Regular; 3.00 points

Comments

N/A

Prerequisites

Linear algebra including Jordan forms and elementary group theory. Some basics of multivariate calculus are needed as well.

Restrictions

Participants:

20

Language of Instruction

English

Attendance and participation

Expected and Recommended

Grade Type

Numerical (out of 100)

Grade Breakdown (in %)

Assignments:

50%

Final:

50%

Evaluation Type

Take-home exam

Scheduled date 1

Date / due date

N/A

Location

N/A

Time

-

Remarks

N/A

Estimated Weekly Independent Workload (in hours)

5

Syllabus

1. Introduction. Examples. Topological groups. GL(n,R) as a topological group. 2. Smooth manifolds. Smooth maps. Direct product. Definition of a Lie group. 3. Tangent space. Immersions and embeddings. Lie subgroup. 4. Elementary properties of Lie groups. Action of a Lie group on a manifold. Representation of a Lie group. Factor group. 5. Closed linear groups. Lie algebra of a closed linear group. 6. Classical groups. 7. Closed Lie subgroups. Factor modulo a closed Lie subgroup. 8. Covering spaces. Fundamental group. Universal covering spaces. Simply connected Lie groups. 9. Vector fields on a manifold. Lie bracket on vector fields. Lie algebra of a Lie group. Closed subgroups of a Lie group (Cartan theorem). 10. Fundamental Lie theorems. 11. Lie groups versus Lie algebras. Commutative groups, center, normal subgroups. Direct product, semidirect product. 12. Compact Lie groups. Representations. Peter-Weyl theorem.

Learning Outcomes

Upon successful completion of this course students should be able to demonstrate an understanding of the basic results of the theory of Lie groups and their representations.

Reading List

1. Lecture notes, will be updated during the course. 2. A. Kirillov, An introduction to Lie groups and Lie algebras. 3. A. Onishchik, E. Vinberg, Lie groups and algebraic groups. 4. V. Serganova et al., Lie groups, Course 261 A, lecture notes.

Website

N/A