Course details
Semester
- Spring 2025
- Monday, January 27th to Friday, April 4th
Hours
- Live lecture hours
- 20
- Recorded lecture hours
- 0
- Total advised study hours
- 80
Timetable
- Thursdays
- 10:05 - 10:55 (UK)
- Fridays
- 10:05 - 10:55 (UK)
Course forum
Visit the https://maths-magic.ac.uk/forums/magic008-lie-groups-and-lie-algebras
Description
Prerequisites
I suppose that my students have certain background in the following topics:
- Linear Algebra
- Abstract Algebra
- Calculus
- Differential Equations
- Differential Geometry and Topology.
- Abstract Algebra: field, group, subgroup, homomorphism, quotient group, cosets, fundamental homomorphism theorem
- Topology: topological and metric spaces, continuous map, homeomorphism, open and closed sets, compactness, connectedness
- Differential Geometry: smooth manifolds, tangent vectors and tangent spaces, smooth maps, differential of a smooth map, vector fields, geodesics, implicit function theorem, submanifolds.
As more or less standard text-books I would recommend:
- John B. Fraleigh, Victor J. Katz, A First Course in Abstract Algebra, 7th edition, Addison-Wesley Publishing, 2002.
- M.A.Armstrong, Basic Topology Undergraduate Texts in Mathematics, 5th printing, Springer, 1997.
- W.A.Sutherland, Introduction to metric and topological spaces, Oxford University Press, 1975.
- B. O'Neill, Elementary Differential Geometry, Harcourt 2nd ed., 1997.
- M. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall Inc., Englewood Cliffs, N.J., 1976.
In general, any feedback from you is very important, since at lectures we will not be able to contact in usual way.
The lectures will be followed by Problem Sheets which will appear regularly on the MAGIC website.
Syllabus
- Manifolds, vector fields, tangent bundle, smooth maps and diffeomorphisms
- Lie groups and Lie algebras, relationship between them
- Classical Lie groups GL(n,R), SL(n,R), O(n), O(p,q), U(n), SU(n), Sp(n,R) and their Lie algebras
- Left and right invariant vector fields, one-parameter subgroups and exponential map
- Fundamental group and universal covering of a Lie group
- Actions of Lie groups
- Homogeneous spaces and linear representations
- Adjoint and co-adjoint representations
- Solvable and nilpotent Lie groups, Lie and Engel theorems
- Killing form, Cartan subalgebra, radical
- Semisimple Lie algebras, classification (without detailed proof)
- Basic facts on root systems and Dynkin diagrams
Lecturer
-
AB
Dr Alexey Bolsinov
- University
- Loughborough University
Bibliography
No bibliography has been specified for this course.
Assessment
The assessment for this course will be released on Tuesday 22nd April 2025 at 00:00 and is due in before Monday 5th May 2025 at 11:00.
Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).
You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).
If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.
Please note that you are not registered for assessment on this course.
Files
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Lectures
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