Introduction to Lie Groups

Calendar

LEC #	TOPICS
1	Historical Background and Informal Introduction to Lie Theory
2	Differentiable Manifolds, Differentiable Functions, Vector Fields, Tangent Spaces
3	Tangent Spaces; Mappings and Coordinate Representation Submanifolds
4	Affine Connections Parallelism; Geodesics Covariant Derivative
5	Normal Coordinates Exponential Mapping
6	Definition of Lie Groups Left-invariant Vector Fields Lie Algebras Universal Enveloping Algebra
7	Left-invariant Affine Connections The Exponential Mapping Taylor’s Formula in a Lie Group Formulation The Group GL (n, R)
8	Further Analysis of the Universal Enveloping Algebra Explicit Construction of a Lie Group (locally) from its Lie Algebra Exponentials and Brackets
9	Lie Subgroups and Lie Subalgebras Closer Subgroups
10	Lie Algebras of some Classical Groups Closed Subgroups and Topological Lie Subgroups
11	Lie Transformation Groups A Proof of Lie’s Theorem
12	Homogeneous Spaces as Manifolds The Adjoint Group and the Adjoint Representation
13	Examples Homomorphisms and their Kernels and Ranges
14	Examples Non-Euclidean Geometry The Associated Lie Groups of Su (1, 1) and Interpretation of the Corresponding Coset Spaces
15	The Killing Form Semisimple Lie Groups
16	Compact Semisimple Lie Groups Weyl’s Theorem proved using Riemannian Geometry
17	The Universal Covering Group
18	Semi-direct Products The Automorphism Group as a Lie Group
19	Solvable Lie Algebras The Levi Decomposition Global Construction of a Lie Group with a given Lie Algebra
20	Differential 1-forms The Tensor Algebra and the Exterior Algebra
21	Exterior Differentiation and Effect of Mappings Cartan’s Proof of Lie’s Third Theorem
22	Integration of Forms Haar Measure and Invariant Integration on Homogeneous Spaces
23	Maurer-Cartan Forms The Haar Measure in Canonical Coordinates
24	Invariant Forms and Harmonic Forms Hodge’s Theorem
25	Real Forms Compact Real Forms, Construction and Significance
26	The Classical Groups and the Classification of Simple Lie Algebras, Real and Complex