Quantum deformation of two four-dimensional spin foam models

Article Outline

1. INTRODUCTION
   1. Background and motivation
   2. Main results
   3. Outline of the paper
2. THE LORENTZIAN MODEL
   1. The quantum Lorentz group
      1. Hopf algebra structure
         1. The Hopf algebra U_q((2)) :
         2. Representation theory of U_q((2)) :
         3. The Hopf algebra F _q (SU(2)):
         4. The quantum Lorentz group D(U_q((2))) :
      2. Irreducible representations, duals, and R -matrix
         1. Irreducible representations of the quantum Lorentz group:
         2. Algebra of functions on the quantum Lorentz group:
         3. Universal R -matrix and braiding:
   2. The quantum EPRL intertwiner
      1. Quantum EPRL representations
      2. Quantum EPRL intertwiners
   3. The four-simplex amplitude
      1. EPRL tensors and bilinear form on the representation spaces
      2. Graphical calculus
      3. Amplitude for the four-simplexes
   4. The quantum spin foam model
3. THE EUCLIDEAN MODEL
   1. The quantum rotation group
      1. U_q((2)) at root of unity
      2. Representation theory of Uq(res)((2)) at a root of unity
         1. Monoidal structure:
         2. Semisimplicity:
         3. Braiding:
         4. Pivotal structure and duals:
         5. Ribbon structure and quantum trace:
         6. Identification of objects and their duals:
      3. Quantum SO(4)
   2. The quantum EPRL intertwiner
   3. Four-simplex amplitude and the quantum spin foam model
4. DISCUSSION AND CONCLUSIONS
   1. Summary
   2. Physical interpretation
   3. Open questions
      1. Note added in proof.