Group Theory

Playlist 1.1 Matrix functions
Matrices : 2 kinds (10 min) 1.2 Matrix functions
Derivative of a matrix function (4 min) 1.3 Matrix functions
Exponential, logarithm of a matrix (7 min) 1.4 Matrix functions
Determinant is a volume (8 min) 1.5 Matrix functions
log det = tr log (8 min) 1.6 Matrix functions
Multi-matrix functions (10 min) 1.7 Matrix functions
Linear differential equations (9 min) 1.8 Eigenvalues and eigenvectors
Projection operators (14 min) 1.9 Eigenvalues and eigenvectors
What's the deal with Hamilton-Cayley? (4 min) 1.10 Eigenvalues and eigenvectors
Spectral decomposition (4 min) 1.11 Eigenvalues and eigenvectors
Spectral decomposition and completeness (4 min) 1.12 Eigenvalues and eigenvectors
Right, left eigenvectors (6 min) 1.13 Eigenvalues and eigenvectors
A projection operators workout (9 min)

Playlist Dresselhaus et al Chapter 1 Basic Mathematical Background Cvitanović et al Chapter 10. Flips, slides and turns Discrete symmetry, an example:
3-disk pinball (4:03 min) What is a group? (10:56 min) Socratica
group multiplication (or Cayley) tables (7:32) Active, passive coordinate transformations (3:08 min) Following Mefisto:
symmetry defined three (3) times (7:01 min) Subgroups, classes, group orbits (7:57 min)

Playlist Dresselhaus et al Chapter 2 Representation Theory and Basic Theorems This requires character (1:23 min) Hard work builds character (15:05 min)
The symmetry group of a propeller (6:13 min)
Irreps of C_3 (14:52 min)
Rotation and reflections in a plane: irreps of D_3 (13:38 min)

Playlist Dresselhaus et al Sect. 2.4 The Unitarity of Representations Dresselhaus et al Sects. 2.5 and 2.6 Schur's Lemma Lecture 3 (Unedited)
Unitarity of reps; Schur's Lemma (2:04:56 h)

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Playlist Dresselhaus et al Sect. 2.7 `Wonderful' Orthogonality Theorem, sect. 2.8 Representations and vector spaces 4.2 Whence "orthogonality"?
The ideas: observables are Hermitian; matrix reps are unitary; average over the group to extract invariants. A matrix rep forms a complex unit vector, hence "orthogonality". (9:47 min) 4.3 Character orthogonality theorem (XX min) Dresselhaus et al Sect. 3.1 Characters and Class to sect. 3.5 The number of irreducible representations 4.4 Characters. Character orthogonality (XX min) 4.5 Class (17:56 min) 4.6 Number of classes equals the number of irreps (6:24 min)

Irreps (3:09 min) Frobenius character formula (12:04 min) Character orthogonality relations (10:53 min) A summary
it is all about class and character (18:50 min)

Playlist Lecture 5 (Unedited) (2:42:51 h)

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Playlist 6.1 A character table
is a unitary matrix from classes to irreps (9 min) 6.2 Projection operator perspective
Ends with the D_3 class multiplication table (12 min) 6.3 Example
The projection operator reduction of D_3 (23 min) 6.4 Example
A reduction by two commuting operators (Harter Problem 1.2.6). What comes next: a nonlinear symmetry reduction; translations; Fourier series. (35 min)

Playlist 7.2 Example
A reduction by two commuting operators, for professionals. Projection operator analysis (xx min) 7.3 Example
The projection operator reduction of D_3 (23 min) Lecture 7 (Unedited) (2:07:51 min)

Playlist Lecture 8 (Unedited) Go nonlinear. Equivariance of Lorenz. Reduction of symmetries of nonlinear equations. (up to 2:42:51 h, stop there) Lecture 9 (Unedited) (2:37:59 h) Lecture 10 (Unedited) (2:29:20 h)

Lecture 12 Lie groups, algebras (Unedited) (2:20:50 min)

Lecture 13 O(2) symmetry sliced (Unedited) (2:13:38 min)

Playlist Lecture 11 (Unedited) (2:20:40 min) 8.1 Example: all SO(3)
A finite 3D rotation (A. Scherbakov) (5 min) 8.2 Example
Infinitesimal rotations (A. Scherbakov) (18 min) 8.3 Example
Commutators (A. Scherbakov) (11 min) 8.4 Example
Commutating operators (A. Scherbakov) (14 min) 8.5 Example
Algebra of angular momenta (A. Scherbakov) (7 min) 8.6 Example
(extra) Eigen-values, -vectors workout (8 min)

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Lecture 12 Lie groups, algebras (Unedited) (2:20:50 min)

Lecture 13 (Unedited) (XX:49 min)

Lecture 14 (Unedited) (2:16:35 h)

Lecture 15 (Unedited) (2:06:49 h)

Lecture 16.3 Main properties of root systems (19:02 min) Lecture 16.4 Construction of Lie algebra representations (21:30 min)

Lecture 17 (Unedited) (2:03:14 h)

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