NOC:An Introduction to Matrix Lie Groups

Lecture 1 - Complex Numbers

Lecture 2 - The Plane Rotations

Lecture 3 - The Quaternions

Lecture 4 - Quaternionic matrices

Lecture 5 - What is a Matrix Lie Group

Lecture 6 - General Linear Group

Lecture 7 - Complex matrices as real matrices

Lecture 8 - Quaternionic matrices as real matrices

Lecture 9 - Discussions on quaternions

Lecture 10 - Topology of matrix groups

Lecture 11 - Topology of matrix groups: Closedness

Lecture 12 - Path-Connected subsets in R^m

Lecture 13 - Compactness in R^m

Lecture 14 - Inner products

Lecture 15 - Unitary transformations

Lecture 16 - Unitary matrices

Lecture 17 - Congruence

Lecture 18 - Euclidean isometries

Lecture 19 - Two dimensional Euclidean Isometries

Lecture 20 - Compactness of unitary groups

Lecture 21 - Connectedness of complex matrix groups

Lecture 22 - Connectedness of real matrix groups

Lecture 23 - Differentiability of a function - I

Lecture 24 - Differentiability of a function - II

Lecture 25 - What is a Lie Algebra

Lecture 26 - Examples of Lie algebras

Lecture 27 - Lie algebra for special linear groups

Lecture 28 - Lie algebra as vector fields

Lecture 29 - Lie algebras for orthogonal and unitary groups

Lecture 30 - Series of Matrices

Lecture 31 - Series of Matrices

Lecture 32 - Exponential map

Lecture 33 - Properties of Exponential Map

Lecture 34 - One parameter Subgroups

Lecture 35 - Matrix logarithm

Lecture 36 - Inverse Function Theorem

Lecture 37 - Image of the Exponential map

Lecture 38 - Exponential map is a local diffeomorphism

Lecture 39 - Every matrix group is a manifold

Lecture 40 - Calculus on Manifolds

Lecture 41 - The Lie Bracket

Lecture 42 - Lie Algebra isomorphisms and Matrix Group isomorphisms

Lecture 43 - Example of isomorphic Lie algebras

Lecture 44 - The adjoint Representation

Lecture 45 - Adjoint representation for compact matrix groups

Lecture 46 - Shape of space rotations

Lecture 47 - More double covers - I

Lecture 48 - More double covers - II

Lecture 49 - Overview of Simply Connectedness

Lecture 50 - Correspondence between connected subgroups and Lie Ideals

Lecture 51 - Discrete subgroups - I

Lecture 52 - Discrete subgroups - II

Lecture 53 - Characterization of Torus

Lecture 54 - Maximal Torus

Lecture 55 - Non-compact Examples: Lorentzian groups

Lecture 56 - Symplectic groups

Lecture 57 - Introduction to Smooth Manifolds

Lecture 58 - Tangent spaces to Smooth Manifolds

Lecture 59 - Lie groups