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://mathsmagic.ac.uk/forums/magic008liegroupsandliealgebras
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 textbooks I would recommend:
 John B. Fraleigh, Victor J. Katz, A First Course in Abstract Algebra, 7th edition, AddisonWesley 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, oneparameter subgroups and exponential map
 Fundamental group and universal covering of a Lie group
 Actions of Lie groups
 Homogeneous spaces and linear representations
 Adjoint and coadjoint 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 Friday 2nd May 2025 at 11:00.
Assessment for all MAGIC courses is via takehome 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 uptodate with the course, the expectation is it should take at most 3 hoursâ€™ 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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