M556 - Lie Groups and Lie Algebras - I

Course No: 
M205, M304, M306
General Properties: Definition of Lie groups, subgroups, cosets, group actions on manifolds, homogeneous spaces, classical groups. Exponential and logarithmic maps, Adjoint representation, Lie bracket, Lie algebras, subalgebras, ideals, stabilizers, center Baker-Campbell-Hausdorff formula, Lie’s Theorems. Structure Theory of Lie Algebras: Solvable and nilpotent Lie algebras (with Lie/Engel theorems), semisimple and reductive algebras, invariant bilinear forms, Killing form, Cartan criteria, Jordan decomposition. Complex semisimple Lie algebras, Toral subalgebras, Cartan subalgebras,Root decomposition and root systems. Weight decomposition, characters, highest weight representations, Verma modules, Classification of irreducible finite-dimensional representations, BGG resolution, Weyl character formula.
Reference Books: 
  1. D. Bump, “Lie Groups”, Graduate Texts in Mathematics 225, Springer, 2013.
  2. J. Faraut, “Analysis on Lie Groups”, Cambridge Studies in Advanced Mathematics 110, Cambridge University Press, 2008.
  3. B. C. Hall, “Lie Groups, Lie algebras and Representations”, Graduate Texts in Mathematics 222, Springer-Verlag, 2003.
  4. W. Fulton, J. Harris, “Representation Theory: A first course”, Springer-Verlag, 1991.
  5. J. E. Humphreys, “Introduction to Lie Algebras and Representation Theory”, Graduate Texts in Mathematics 9, Springer-Verlag, 1978.
  6. A. Kirillov, “Introduction to Lie Groups and Lie Algebras”, Cambridge Studies in Advanced Mathematics 113, Cambridge University Press, 2008.

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