Theory of Fundamental Interactions (Lecture)

Problem sheets

The number in brackets refers to the calendar week in which the corresponding sheet will be discussed in the exercise class.

• Sheet 1 (17): problems and solutions
   
  
  1. Adjoint representation of (ℝ³,×).
  2. ad is a Lie-algebra homomorphisms.
  3. ad acts by isometries with respect to Killing-form.
  4. Killing-form for Lie(SO(3)).
  5. Semi-direkt product of Lie-algebras.
  6. Proof of simplicity of Lie(SL(2,ℝ)).
  7. ℂ ⊗L of a real Lie-algebra L; preservation of semi-simplicity and compacness?
  8. Isomorphicity of ℂ⊗Lie(SU(2)) and ℂ⊗Lie(SL(2,ℝ)).
   

• Sheet 2 (19):  problems and solutions
   
  
  1. Pauli matrices.
  2. Uniqueness of direct sum decomposition of semi-simple Lie algebra into simple ones.
  3. Exponetial of complex linear combination of Pauli matrices.
  4. Simple and semi-simple Lie algebras are perfect. What about the corresponding inhomogeneous groups?
  5. SL(2,ℝ) has no non-trivial finite-dimensional representations.
  6. Lie algebras of generalised orthogonal groups.
  7. Adjoint representations of inhomogeneous Lie groups and algebras.
  8. Co-adjoint representation of inhomogeneous Lie groups.
   

• Sheet 3 (22):  problems and solutions
   
  
  1. Spatial rotations in SU(2).
  2. Parametrisation of most general SU(2) matrix.
  3. The generalised orthogonal group O(V,ω) its Lie-algebra.
   

• Sheet 4 (25):  problems and solutions
   
  
  1. Covering homomorphism SL(2,ℂ) → Lor_+↑ (proper orthochronous Lorentz group).
  2. Explicit check: Lie algebras of inhomogeneous generalised orthogonal groups are perfect; implication for one-dimensional representations. Also: behaviour under change of origin.
  3. Representation of inhomogeneous generalised orthogonal groups and their Lie-algebras on function spaces.
  4. Lie-algebras of the deSitter and anti-deSitter groups and some of their contractions.
  5. How to define the worldline for "centre-of-mass" in Poincaré invariant mechanics. (A problem for enthusiasts!)

• Sheet 5 (27):  problems and solutions
   
  
  1. Irreducible SO(3)-subspaces in ℝ³ ⊗ ℝ³. 
  2. SO(3)-irresucibility of space of totally symmetric traceless tensors over ℝ³.
  3. Representation of SO(3) and Lie(SO(3)) on C^∞(S²,ℂ) and construction of spherical harmonics Y_ℓm by ladder operators.
  4. Decomposition of totally symmetric valence-n spinor into symmetric tensor product of n spinors.
  5. Real structures on direct-sum and tensor-product spaces of a complex vector space.
  6. Complexification of a real Lie-algebra L decomposes into direct sum of ideals if L already has a complex structure.