Special Mathematics Lecture
Contact:
Serge Richard (richard@math.nagoyau.ac.jp), Rm. 247 in Sci. Bldg. A
Groups and their representations (Fall 2022)
Registration code : 0063621
Schedule : Wednesday (18.30  20.00) in room 207 of Science building A

Class dates :
October 5, 12, 19, 26
November 2, 9, 16, 23, 30
December 7, 14, 21
January 11, 18

Program :
1) Groups
2) Linear representations
3) Lie groups and Lie algebras
4) Semisimple theory
5) Examples

Weekly summaries :
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14

Study sessions :
Are organized on an individual basis by some students.
For any information, contact
Duc Dam,
Tom,
Qi,
Vic,
Jongruk.

For the evaluation, you need to submit the solutions of some exercises and/or the proofs of some statements.
These submissions can take place at any time during the semester.
If you have any question, contact me
or Duc Dam.

Works submitted by the students :
Normal subgroups and quotient groups, by Li Yucheng
On the center of a group, by Rafi Rizqy Firdaus
On inner and outer semidirect products, by Vic Austen
On groups and normal subgroups, by Haruki Tsunekawa
Center and homomorphisms, by Haruki Tsunekawa
About O(n) and SO(n), by Haruki Tsunekawa
On quotient groups, by Magnus B. Lyngby
On irreducible representations of finite groups, by Li Yucheng
Normal subgroups, center, and morphisms, by Ai Yamada
Unicity in a group, and conjugation classes, by Rafi Rizqy Firdaus
On faithful representations, by Nguyen Quan Minh
The Poincaré group, a semidirect product, by Rafi Rizqy Firdaus and Jongruk Pukdee
Orthogonality of characters, by Nguyen Tue Tai
Proof of Schur's Lemma, by Vic Austen
On the regular representation for finite group, by Nguyen Quan Minh
Projective rays and pure states, by Li Yucheng
Existence of unitary representations for finite groups, by Rafi Rizqy Firdaus
Trace and norm of the tensor product of linear operators, by Nguyen Tue Tai
Trace of a tensor product operator, by Haruki Tsunekawa
Topological continuity and epsilon  delta argument , by Rafi Rizqy Firdaus
Commutator of matrices, and Lie algebras , by Emika Sekiya
Proof of the selection rule, by Li Yucheng
NonHausdorff topological spaces, and Hausdorff implies T1, by Rafi Rizqy Firdaus
The center of any Lie algebra is an ideal, by Sirawich Saranakomkoop
On the adjoint representation for semisimple and simple Lie algebras, by Li Yucheng
Weights and weight vectors of an irreducible representation, by Li Yucheng
About the adjoint representation of a Lie algebra , by Emika Sekiya and Peng Qi
Lie bracket and structure coefficients, by Hadiko Rifqi Aufa Sholih
Open and closed sets in topological spaces, by Matefy Adam
Complexification of a Lie algebra, by Ai and Erin Yamada
A Proof of Maschke's theorem on vector space, by Takami Sugimoto
On the second countability of R and R^n, by Magnus B. Lyngby
On compactness, by Pratham Prashant Dhomne and Haruki Tsunekawa
Definition of Lie Group and proof that the identity component is a normal subgroup, by Magnus B. Lyngby
About the induced representation, by Nguyen Tue Tai
The restricted Lorentz group and notions of orthochronous proper Lorentz transformations, by Cole Vincent
Stabilizers and orbits, by Cole Vincent
BakerCampbellHausdorff Theorem , by Nguyen Thanh and Peng Qi
Restricted Lorentz Group and Lorentz Invariance in Physics, by Rafi Rizqy Firdaus
The Euclidean group and the Poincaré group as Lie groups, by Hevidu Kanishka Samarakoon
A report on some exercises from Chapter 1, by Matefy Adam
Regular representation and tensor product of Hilbert spaces, by Yesui Baatar
SU(2) to SO(3) homomorphism, by Nguyen Duc Thanh and Yesui Baatar
Crystallographic Groups, by Nguyen Duc Thanh
On the WignerEckart Theorem, by Nguyen Duc Thanh
Heisenberg Group, by Ryotaro Arakawa

References : (electronic version available upon request)
[A] Lecture notes: Theorie des groupes pour la physique, written by W. Amrein.
Excellent, but in French
[B] Book: An introduction to differentiable manifolds and Riemannian geometry, written by W. Boothby.
A classical and beautiful book on differential geometry
[C] Book: Group theory in physics, an introduction, written by J.F. Cornwell.
Looks appropriate. Based on previous volumes I and II
[F] Book: A course in abstract harmonic analysis, written by G. Folland.
Excellent, but clearly more advanced
[Ge] Book: Lie algebras in particle physics, from isospin to unified theories, written by H. Georgi.
Written for physicists, but provides a good introduction with many heuristic explanations
[Gu] Lecture notes: Group theory
and its applications in physics, written by B. Gutkin.
Much more physically oriented
[H] Book: Lie groups, Lie algebras, and representations: an elementary introduction, written by B. Hall.
Almost everything on linear Lie groups and algebras, quite accessible
[Le] Lecture notes: Notes on Lie Groups, written by E. Lerman.
Covers the main topics, but it is much longer
[Lu] Lectures notes: Group theory (for physicists), written by C. Ludeling.
Covers the main topics
[O] Lecture notes: Symmetries and groups, written by H. Osborn.
More advanced, with material on gauge groups and gauge theories
[RS] Book: Functional analysis I, written M. Reed and B. Simon.
A classical book of functional analysis
[R] Book: A course in the theory of groups, written by D.J. Robinson.
More complicated and more complete, notations not so friendly
[Sc] Lecture Notes: Group Theory Summary, written by M. Schaller.
Very brief text
[SS] Book: Linear representations of finite groups, written by J.P. Serre and L. Scott.
Very exhaustive but for finite groups only. Contains some simple examples
[Si] Book: Representations of finite and compact groups, written by B. Simon.
Excellent, but little bit advanced
[St] Book: Group theory and physics, written by S. Sternberg.
Looks quite accessible
[Z] Lecture notes: Invariances in Physics
and Group Theory, written by J.B. Zuber.
More advanced, with material on gauge theories
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