# Topological Matrix Groups (MAE826)

### General

School School of Science Department of Mathematics Undergraduate MAE826 8 Topological Matrix Groups Interactive, Presentations (Weekly Teaching Hours: 3, Credits: 6) Special Background, skills development - Greek, English Yes See eCourse, the Learning Management System maintained by the University of Ioannina.

### Learning Outcomes

Learning outcomes The aim of the course is to provide an introduction to Lie theory through matrix groups. The main subject of study is the closed subgroups of the general linear group. Our study is extended from real to complex and quaternion numbers. The corresponding linear groups are in fact topological groups and an introduction of basic properties of topological group is also provided. The Lie algebra of a matrix group is defined. The special orthogonal, unitary and symplectic groups provide important example of Lie algebras. Lie algebras are studied using the exponential map. Finally Lie groups are defined. Study particular characteristics of group theory in topology and geometry Independent and team work Working in an interdisciplinary.

### Syllabus

• General linear groups
• Real and Complex algebras, Quaternions. Matrix algebras
• Inner product, orthogonal, unitary and symplectic groups
• Homomorphisms
• Differential curves, tangent vectors. Dimension of a matrix group
• Differential homomorphisms
• Expontential and logarithmic funcions. Lie algebras
• Special orthogonal and symplectic groups
• Topological groups, manifolds
• Maximal tori
• Differential manifolds, Lie groups.

### Teaching and Learning Methods - Evaluation

Delivery

Face-to-face, Distance learning

Use of Information and Communications Technology

Communication with students

Teaching Methods
Lectures 39
Working hours in class 8
Project 30
Assignments 33
Final exam 41
Course total 150
Student Performance Evaluation

Written Examination, Oral Presentation, written assignments.

### Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• J. F. Adams, Lectures on Lie groups, University of Chicago Press, 1969.
• M. L. Curtis, Matrix Groups, Springer-Verlag, 1979.
• R. Howe. Very basic Lie theory, American math. monthly,90, 1983.