Course Details:

Recalling Differentiable manifolds & Topological groups. Lie groups and their properties. Matrix groups as examples. Cosets, group actions on manifolds, homogeneous spaces. Lie algebra of a Lie group. Lie algebra of a Lie group including Lie-Trotters formula. Cartan’s theorem and applications. Adjoint representation and their properties. Uniqueness of Differential structure. Derived representation and their properties. Solvable and nilpotent Lie algebras (with Lie/Engel theorems), Reductive algebras, Killing form, Cartan criteria Jordan decomposition, complex semisimple Lie algebras, complex semisimple Lie algebras, Nilpotent & solvable Lie groups.

Text Books:

  1. Hilgert & NeebStructure and Geometry of Lie groupsSpringer, 2010
  2. J. Faraut: Analysis on Lie groupsCambridge studies in Advanced mathematics 110, 2008

Reference Books:

  1. D. Bump, Lie groups, Springer, 2004.
  2. Brian C. Hall, Lie groups, lie algebras, and representations: An elementary introduction.
  3. W. Fulton & J. Harris, Representation theory: A first course, Springer - Verlag, 1991.
  4. J. E. Humpreys, Introduction to lie algebras and representation theory, Springer, 1972.
  5. V.S. Varadharajan, Lie groups, lie algebras, and their representations, Springer, 1984.
  6. F. Warner, Foundations of differentiable manifolds and lie groups, Springer, 1984.