The theory of Lie groups and algebras is a fundamental tool in many diverse areas of mathematics such as
analysis, differential geometry, differential equations, number theory, coding etc.. Their applications to
other areas of sciences such as atomic and molecular structures, high energy particle physics, solid states
physics, etc. are also well-known. This course is an introduction to the theory of Lie groups and algebras
along with some sampling of their applications. We will devote most of the time to the representation
theory of the compact Lie groups and the semi-simple Lie algebras.
The main contents are listed below:
(a) Notations and terminologies from manifold theory. Lie groups and their algebras. Matrix Lie groups:
GL(n), SL(n) ,O(n), SO(n), U(n), Sp(n) , E(n), and Heisenberg group. Exponential mapping. The
Trotter product formula. Baker-Campbell-Hausdorff formula. Closed subgroups theorem. Covering
groups and simply-connected Lie groups. Lie groups and Lie algebras correspondence.
(b) Adjoint representation of a Lie group. adjoint representation of a Lie algebra. Solvable, nilpotent and
simple Lie algebras. Engel's and Lie's theorems. Killing form and sem-=simplicity. Cartan's
criterion. Basic representation theory of a Lie algebra. sl(2, R) and 81(2, C). Weyl's theorem.
80(3, R) and su(2).
(c) Representation of a Lie group. Reducible and irreducible representations. Schur's Lemma. Haar
measure. Unitary representations. Orthogonality of distinct irreducible representations of a compact
Lie group. Characters. Complete reducibility of a representation of a compact Lie group.
Representations of a compact connected Abelian Lie group (i.e torus). Maximal tori in compact Lie
group. Weyl groups and chambers. Peter-Weyl theorem. The Boreil-Weyl construction. Groups vs
Lie algebra representations.
(d) Complete reducibility and semi-simple Lie algebras. Cartan subalgebras. Roots and root systems.
Weyl groups and chambers. Classification of irreducible root systems. Cartan matrices and Dynldn
diagrams. Serre's theorem.
(e) Weights and highest weight representations. Universal enveloping algebras. Casimir operators.
Poincar&Birkhoff-Witt theorem. Verma modules. Weyl character and dimension formulas.
Clesbsch-Gordan coefficients. Compact Lie algebras. Representations of classical compact matrix Lie
groups.
(f)* General Cartan matrices. Affine Lie algebras. Central extensions and Virasoro algebras.
Representations of affine sZ(2, C) and KdV hierarchy.
The subject (f) is optional if time is permitted.