Lie Groups and Lie Algebra

Linear Lie Groups: the exponential map and the Lie algebra of linear Lie group, some calculus on a linear Lie group, invariant differential operators, finite dimensional representations of a linear Lie group and its Lie algebra. Examples of linear Lie group and their Lie algebras, e.g., Complex groups: GL(n, C), SL(n, C), SO(n, C), Groups of real39matrices in those complex groups: GL(n, R), SL(n, R), SO(n, R), Isometry groups of Hermitian forms SO(m, n),U(m, n), SU(m, n). Finite dimensional representations of su(2)and SU(2) and their connection. Exhaustion using the lie algebra su(2). Lie algebras in general, Nilpotent, solvable, semisimple Lie algebra, ideals, Killing form, Lies and Engels theorem. Universal enveloping algebra and Poincare-Birkho-Witt Theorem (without proof). Semisimple Lie algebra and structure theory: Definition of Linear reductive and linear semisimple groups. Examples of Linear connected semisimple/ reductive Lie groups along with their Lie algebras (look back at 2 above and find out which are reductive/ semisimple).Cartan involution and its differential at identity; Cartan decomposition g = k + p, examples of k and p for the groups discussed above. Definition of simple and semisimple Lie algebras and their relation, Cartans criterion for semisimplicity. Statements and examples of Global Cartan decomposition, Root space decomposition; Iwasawa decomposition; Bruhat decomposition.