Lie groups and Lie algebras (MAGIC008)
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This course is part of the MAGIC core.
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
Autumn 2018 (Monday, October 8 to Friday, December 14)
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:
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. Alexey Bolsinov
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
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Assessment for this course will be via a take-home examination which will be put online after the end of the course at the beginning of January (morning, 6th January 2019). 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, 20th January 2019.
No assignments have been set for this course.
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