  A family of infinite-dimensional irreducible $\star$-representations on
$\mathcal{H}\simeq L^2(\mathbb{R})\otimes\mathbb{C}^k$ is defined for a
quantum-deformed Lorentz algebra $U_\mathbf{q}(sl_2)\otimes
U_{\tilde{\mathbf{q}}}(sl_2)$, where $\mathbf{q}=\exp[\frac{2\pi i}{k}(1+b^2)]$
and $\tilde{\mathbf{q}}=\exp[\frac{2\pi i}{k}(1+b^{-2})]$ with
$k\in\mathbb{Z}_+$ 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 $\mathrm{SL}(2,\mathbb{C})$ 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
$\mathrm{SL}(2,\mathbb{C})$ 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 $\mathcal{H}$, which
we call the Fenchel-Nielson representation.
