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Course11110690459 lectures

An Introduction to Matrix Lie Groups

Prof. Krishnendu Gongopadhyay · IISER Mohali

Start with lecture 1 At NPTEL

Lectures

Week 1

  1. Complex Numbers Video
  2. The Plane Rotations Video
  3. The Quaternions Video
  4. Quaternionic matrices Video

Week 2

  1. What is a Matrix Lie Group Video
  2. General Linear Group Video
  3. Complex matrices as real matrices Video
  4. Quaternionic matrices as real matrices Video
  5. Discussions on quaternions Video

Week 3

  1. Topology of matrix groups Video
  2. Topology of matrix groups: Closedness Video
  3. Path-Connected subsets in R^m Video
  4. Compactness in R^m Video

Week 4

  1. Inner products Video
  2. Unitary transformations Video
  3. Unitary matrices Video
  4. Congruence Video
  5. Euclidean isometries Video
  6. Two dimensional Euclidean Isometries Video

Week 5

  1. Compactness of unitary groups Video
  2. Connectedness of complex matrix groups Video
  3. Connectedness of real matrix groups Video
  4. Differentiability of a function Video
  5. Differentiability of a function-II Video

Week 6

  1. What is a Lie Algebra Video
  2. Examples of Lie algebras Video
  3. Lie algebra for special linear groups Video
  4. Lie algebra as vector fields Video
  5. Lie algebras for orthogonal and unitary groups Video

Week 7

  1. Series of Matrices Video
  2. Series of Matrices Video
  3. Exponential map Video
  4. Properties of Exponential Map Video
  5. One parameter Subgroups Video
  6. Matrix logarithm Video

Week 8

  1. Inverse Function Theorem Video
  2. Image of the Exponential map. Video
  3. Exponential map is a local diffeomorphism Video
  4. Every matrix group is a manifold Video
  5. Calculus on Manifolds Video

Week 9

  1. The Lie Bracket Video
  2. Lie Algebra isomorphisms and Matrix Group isomorphisms Video
  3. Example of isomorphic Lie algebras Video
  4. The adjoint Representation Video
  5. Adjoint representation for compact matrix groups Video

Week 10

  1. Shape of space rotations Video
  2. More double covers Video
  3. More double covers-II Video
  4. Overview of Simply Connectedness Video
  5. Correspondence between connected subgroups and Lie Ideals Video

Week 11

  1. Discrete subgroups - I Video
  2. Discrete subgroups - II Video
  3. Characterization of Torus Video
  4. Maximal Torus Video

Week 12

  1. Non-compact Examples: Lorentzian groups Video
  2. Symplectic groups Video
  3. Introduction to Smooth Manifolds Video
  4. Tangent spaces to Smooth Manifolds Video
  5. Lie groups Video

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IISER Mohali
Department
Mathematics
Teacher
Prof. Krishnendu Gongopadhyay
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
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