Course details
A core MAGIC course
Semester
 Autumn 2020
 Monday, October 5th to Friday, December 11th
Hours
 Live lecture hours
 20
 Recorded lecture hours
 0
 Total advised study hours
 80
Timetable
 Wednesdays
 09:05  09:55
 Fridays
 09:05  09:55
Course forum
Visit the MAGIC008 forum
Description
Lie groups, Lie algebras, classical matrix groups GL(n,R), SO(n), SO(p,q), U(n), Lorentz group, Poincare group; exponential map, oneparameter subgroups; actions and basic representation theory, orbits and invariants; adjoint and coadjoint representations, LiePoisson bracket; solvable, nilpotent and semisimple Lie algebras
Prerequisites
The main goal of my course is to give an introduction to the theory of Lie groups and Lie algebras as well as to discuss some applications of this theory to mathematical physics and mechanics.
I suppose that my students have certain background in the following topics:
I suppose that my students have certain background in the following topics:
 Linear Algebra
 Abstract Algebra
 Calculus
 Differential Equations
 Differential Geometry and Topology.
Since Abstract Algebra, Differential Geometry and Topology are all very essential for this course, let me list some basic notions and results which will be used throughout my course.
 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.
Of course, giving the course (especially in the beginning) I will try to recall all these notions. But unfortunately I am not able to discuss them in detail, so it is strongly recommended to have a look at the corresponding literature to refresh your background.
As more or less standard textbooks I would recommend:
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.
I would be glad to get any comments from you related to the above list: if you think that I should pay particular attention to some of the above notions, I'll try to spend on that more time.
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.
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
Follow the link for a book to take you to the relevant Google Book Search page
You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library'  this sometimes works well, but not always  you will need to enter your location, but it will be saved after you do that for the first time.
 Introduction to lie algebras (Erdmann and Wildon, book)
 Lie groups beyond an introduction (Knapp, book)
 Introduction to Lie algebras and representation theory (Humphreys, book)
 Matrix groups: an introduction to Lie group theory (Baker, book)
 Lie algebras and Lie groups: 1964 lectures given at Harvard University (Serre, book)
 Lectures on Lie groups (Adams, book)
 Lie groups: an introduction through linear groups (Rossmann, book)
 Lie groups, Lie algebras, and representations: an elementary introduction (Hall, book)
Assessment
Description
Assessment for this course will be via a takehome examination which will be put online after the end of the course at the beginning of January (morning, 11th January 2021). The exam paper will require the completion of 3/4 questions and to pass one is required to obtain at least 50 %. The deadline for the work to be completed is midnight, 22th January 2021.
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Files
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Lectures
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