Representations of a quantum-deformed Lorentz algebra, Clebsch-Gordan map, and Fenchel-Nielsen representation of quantum complex flat connections at level-k

A family of infinite-dimensional irreducible -representations on H L² (R) ᵏ is defined for a quantum-deformed Lorentz algebra Uq (sl₂) Uₐ (sl₂), where q=2 ik (1+b²) and q=2 ik (1+b^-2) with k_+ and |b|=1. The representations are constructed with the irreducible representation of quantum torus algebra at level-k, which is developed from the quantization of SL (2, C) Chern-Simons theory. We study the Clebsch-Gordan decomposition of the tensor product representation, and we show that it reduces to the same problem as diagonalizing the complex Fenchel-Nielson length operators in quantizing SL (2, C) flat connections on 4-holed sphere. Finally, the spectral decomposition of the complex Fenchel-Nielson length operators results in the direct-integral representation of the Hilbert space H, which we call the Fenchel-Nielson representation.