Group theory II 2025
Published:
In spring 2025, I took the class “Group theory II” (Lie groups and Lie algebras) by Prof. Yuxin Liu (刘玉鑫). Because of the retirement of a professor, the course schedule this year was adjusted, and this class, originally taught by Prof. Yinan Wang (王一男), was assigned to Prof. Yuxin Liu, who taught this class before 2020. To my opinion, Prof. Liu’s class seems to be better, because it has a broader context, explains everything clearly without involving complex mathematical concepts, and introduces physical applications (especially in nuclear theory) in detail.
Reference books
The main reference book is Prof. Yuxin Liu’s 《物理学家用李群李代数》(Lie groups and Lie algebras for physicists), and I found the book 《物理学中的群论（第二版）》(Group theory for physicists (second edition)) by 马中骐 (Zhongqi Ma) useful. Other references books include:
- Barut and Raczka, Theory of Group Representations and Applications.
- Hamermesh, Group theory and its application to physical problems.
- Wybourne, Classical Groups for Physicists.
- 万哲先 (Zhexian Wan), 李代数 (Lie algebra).
- Iachello, Lie Algebras and Applications.
- Iachello and Arima, The Interacting Boson Model.
Course content
The main content of this course includes:
- Basic concepts of Lie groups and Lie algebras
- Lie groups and Lie transformation groups, infinitesimal generators, basic properties of Lie groups (Lie’s three theorems, connectivity and coverage), Lie algebras
- Bases of Lie algebras, Killing metric, structure of Lie algebras (solvable, nilpotent, simple, semisimple, decomposition of Lie algebras), Casimir operators
- Semisimple Lie algebras
- Canonical form of Lie algebras, Cartan-Weyl basis
- Root system of semisimple Lie algebras, Dynkin diagram
- Determination of roots
- Realization of Lie algebras: Cartan-Weyl basis, Chevalley basis, fermion and boson realizations
- Representation of Lie algebras
- Weight systems
- Irreducible tensors and the Young tableaux
- Representation of classical Lie algebras
- Casimir operators
- Decomposition of representations
- Branching rules of Lie algebras
- Decomposition of direct products of representations, Clebsch-Gordon coefficients and Wigner-Eckart theorem
- Racah theorem and isoscalar factors
- Spacetime symmetry and applications
- Lorentz group and its representations
- Applications of the Lorentz group and the Poincare group
- Applications of Lie algebras to hadron structures and strong interactions
- Quark structure of light-flavor hadrons and many-quark states
- Chiral symmetry and its breaking
- Gauge symmetry
- Applications of Lie algebras to many-particle systems
- Dynamical symmetry of fermionic and bosonic systems
- Collective motion of many-particle systems, coherent states
- Interacting boson model of nuclear systems, spectrum generating algebra, shape phase transitions
- Algebraic models of superconductivity
Essay
We were required to write an essay on any topic about Lie group and Lie algebras. I chose the topic “Branching rules and Gelfand-Tsetlin bases of the classical Lie algebras”.
Branching rules and Gelfand-Tsetlin bases of the classical Lie algebras
Notes
If possible, I will organize my notes into a short book, and upload it to this site.