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)⊗ Uq̃(sl₂), where q=exp[2π i/k(1+b²)] and q̃=exp[2π i/k(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.