MAGIC008: Lie groups and Lie algebras

Lie groups, Lie algebras, classical matrix groups GL(n,R), SO(n), SO(p,q), U(n), Lorentz group, Poincare group; exponential map, one-parameter subgroups; actions and basic representation theory, orbits and invariants; adjoint and coadjoint representations, Lie-Poisson bracket; solvable, nilpotent and semisimple Lie algebras

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: 

•  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 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. 

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. 

1. Manifolds, vector fields, tangent bundle, smooth maps and diffeomorphisms 
2. Lie groups and Lie algebras, relationship between them 
3. 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 
4. Left and right invariant vector fields, one-parameter subgroups and exponential map 
5. Fundamental group and universal covering of a Lie group 
6. Actions of Lie groups 
7. Homogeneous spaces and linear representations 
8. Adjoint and co-adjoint representations 
9. Solvable and nilpotent Lie groups, Lie and Engel theorems 
10. Killing form, Cartan subalgebra, radical 
11. Semisimple Lie algebras, classification (without detailed proof) 
12. Basic facts on root systems and Dynkin diagrams