Group Theory

by G. Mussardo (2 credits, type C)


Rubik_cube

(image credits: Wikipedia)

Main topics

  • Finite Groups
    • 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
  • Lie Groups
    • Generators and Commutators
    • Baker-Haussdorf formula
    • Lie Algebras
    • Simple and semi-simple algebras
    • The adjoint representation
  • The building block of all: SU(2)
    • Raising and lowering operators
    • Irreducible representations
    • Coherent states
    • Tensor products and 6j-symbols
    • Quantum spin chains
    • Cold atoms in a two-well trap
  • Wigner-Eckart theorem
    • Structure theory
      • Cartan algebra
      • Roots and weights
      • Raising and lowering operators
      • Structure constants and normalizations
      • Constructing the Lie algebras
        • 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
        • Unitary Groups
          • Irreducible representations and Young tableaux
          • The classical and exceptional groups
            • Ising model in a magnetic field, glimpse on E8 structure of nature
            • The Classification Theorem of semi-simple Lie algebras
              • Global and gauge symmetries in field theories
                • Two famous bosons: Goldstone and Higgs

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