by G. Mussardo (2 credits, type C)
(image credits: Wikipedia)
Main topics
- Properties
- Representations
- Examples: Platonic Solid, Crystals, Space and Plane Tessellation
- Energy spectrum in periodic and a-periodic systems
- Point Groups
- Permutations: Cayley’s theorem and Young Tableaux
- Potts model and random systems
- Echoes from number theory
- Generators and Commutators
- Baker-Haussdorf formula
- Lie Algebras
- Simple and semi-simple algebras
- The adjoint representation
- Raising and lowering operators
- Irreducible representations
- Coherent states
- Tensor products and 6j-symbols
- Quantum spin chains
- Cold atoms in a two-well trap
- Cartan algebra
- Roots and weights
- Raising and lowering operators
- Structure constants and normalizations
- Positive weights
- Simple roots
- A lot of SU(2)’s groups
- The kaleidoscope of the Weyl group
- The Dynkin diagrams
- Examples from rank 2 algebras:
- 1) SU(3) and eightfold way
- 2) Being exceptional, G2
- Irreducible representations and Young tableaux
- Ising model in a magnetic field, glimpse on E8 structure of nature
- Two famous bosons: Goldstone and Higgs