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Ingo Runkel

Topics in representation theory

Announcements:
  • [11.2] The solution to sheet 8 is now also available below.
  • [30.1] There are errors on exercise sheets 11 and 12: in exercise 49, A should be semisimple and V should be irreducible. In exercise 53, the map phi is missing a minus sign (this sign was also missing in the lecture). Both errors have now been fixed on the online sheets.
Times and rooms:
Lecture Tuesday 12:15-13:45 in H2 and Friday 12:15-13:45 in H3. Excercise class Monday 14:15-15:45 in 432.

Exercise sheets:
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

Solutions:
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

Description:
In this course we will study some topics in representation theory with applications in theoretical physics. We will cover aspects of the following:
  • SU(N) and Schur-Weyl duality (representations of the symmetric groups via Young diagrams, Schur-Weyl duality, representations of SU(N))
  • Poincare algebra (classification of unitary positive energy representations, finite dimensional representations)
  • Clifford algebras (classification of representations over real and complex numbers, Spin-groups)
  • Supersymmetry (supersymmetric extensions of the Poincare algebra, representations)
The course is primarily aimed at Master students in Mathematics and Mathematical Physics.

Prerequisites:
Basic knowledge in algebra (linear algebra, basics of groups, rings and representations) and analysis (Hilbert spaces, linear operators).

Literature:
  • SU(N) and Schur-Weyl duality
    main references:
    • Fulton, Harris, Representation theory, Springer
    • Baker, Matrix groups, Springer
    further references:
    • Anderson, Fuller, Rings and categories of modules, Springer
    • Bump, Lie groups, Springer

  • Unitary representations of the Poincare group
    main references:
    • Schottenloher, A mathematical introduction to conformal field theory, Springer
    • Barut, Theory of group representations and applications, Polish Sci.Publ.
    further references:
    • Bargmann, Note on Wigner's Theorem on Symmetry Operations, J. Math. Phys. 5, 862 (1964)
    • Artin, Geometric Algebra, Interscience publishers, 1957
    • Knapp, Lie groups beyond an introduction, Birkhäuser
    • Straumann, Unitary Representations of the inhomogeneous Lorentz Group and their Significance in Quantum Physics, arXiv:0809.4942

  • Clifford algebras and Spin groups
    main reference:
    • Lawson, Michelson, Spin geometry, Princeton University Press
    further references:
    • Lounesto, Clifford Algebras and Spinors, Cambridge University Press
    • Lam, Introduction To Quadratic Forms Over Fields, AMS
    • Varadarajan, Supersymmetry for Mathematicians, AMS

  • Supersymmetry
    main reference:
    • Varadarajan, Supersymmetry for Mathematicians, AMS
    further references:
    • Freed, Five lectures on supersymmetry, AMS
    • Carmeli, Cassinelli, Toigo, Varadarajan, Unitary representations of super Lie groups and applications to the classification and multiplet structure of super particles, hep-th/0501061


Exam:
There will be an oral exam for this course. To qualify for the exam, you need to solve enough homework problems: At the beginning of each class, you can mark on a list which problems (or problem parts, for longer problems) you solved and can present on the board. To qualify for the exam, you should have marked 40% or more of the problems by the end of the course. If you marked a problem as solved but it becomes apparent that you did not prepare it sufficiently in case you get asked to the board, the all problems for that week will be marked as not solved.
 
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