M564 - Groups and Lie Algebras - II

Course No: 
General theory of representations, operations on representations, irreducible representations, Schur’s lemma, Unitary representations and complete reducibility. Compact Lie groups, Haar measure on compact Lie groups, Schur’s Theorem, characters, Peter-Weyl theorem, universal enveloping algebra, Poincare-Birkoff-Witt theorem, Representations of Lie(SL(2, C)). Abstract root systems, Weyl group, rank 2 root systems, Positive roots, simple roots, weight lattice, root lattice, Weyl chambers, simple reflections, Dynkin diagrams, classification of root systems, Classification of semisimple Lie algebras. Representations of Semisimple Lie algebras, weight decomposition, characters, highest weight representations, Verma modules, Classification of irreducible finite-dimensional representations, Weyl Character formula, The
representation theory of SU (3), Frobenius Reciprocity theorem, Spherical Harmonics.
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. A. Kirillov, “Introduction to Lie Groups and Lie Algebras”, Cambridge Studies in Advanced Mathematics 113, Cambridge University Press, 2008.
  6. A. W. Knapp, “Lie Groups: Beyond an introduction”, Birk ̈ auser, 2002.
  7. B. Simon, “Representations of Finite and Compact Groups”, Graduate Studies in Mathematics 10, American Mathematical Society, 2009.

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