Entry requirements

120 credits in mathematics and/or physics. Participation in Mathematical Methods of Physics II or Differential topology. Symmetry and Group theory in Physics or Algebraic Structures. Analytical mechanics. Complex analysis.

Learning outcomes

On completion of the course, the student should be able to:

• Explain how to use Schur-Weyl duality.
• Construct explicit representations and their characters for classical Lie algebras, provide examples of vector and spinorial representations in physics.
• Explain the classification of affine Kac-Moody algebras.
• Provide examples of Hopf algebras.
• Reformulate the canonical (Hamiltonian) formalism of classical mechanics in the language of symplectic geometry.
• Explain the procedure of quantisation.
• Give examples of classical and quantum integrable systems.
• Use results from representation theory to perform explicit spectrum computations.
• Apply techniques of integrability to selected physical models.

Content

Permutation group and classical Lie groups and algebras. Schur-Weyl duality. Computation techniques to work with vector and spinorial representations. Serre-Chevalley description of Lie algebras. Loop algebras and their central extension. Affine Kac-Moody algebras and their classification. Oscillator realisations. Combining Heisenberg algebra and simple Lie algebras. Coproduct and Hopf algebras.

Hamiltonian formalism in the language of symplectic geometry. Sympletic and Poisson manifolds. Liouville integrability and separation of variables. Hamiltonian reduction. 

Poisson-Lie groups. Deformation quantisation and quantum groups.

Quantum integrability. S-matrix and Yang-Baxter equation, its simplest solutions. RTT realisation of Yangian and quantum affine algebras. Construction of integrable Hamiltonians on the examples of spin chains. Computation of their spectrum using integrability techniques.

Instruction

Lectures, problem-solving and discussion sessions. In case of a small number of participants, the course may be given in the format of reading project.        

Assessment

Hand-in problems during the course.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.

Transitional provisions

The course is given first time in HT2026.