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A<sc>bstract</sc> This paper develops a framework for the Hamiltonian quantization of complex Chern-Simons theory with gauge group$$\text{SL}(2,{\mathbb{C}})$$at an even level$$k\in {\mathbb{Z}}_{+}$$. Our approach follows the procedure of combinatorial quantization to construct the operator algebras of quantum holonomies on 2-surfaces and develop the representation theory. The *-representation of the operator algebra is carried by the infinite dimensional Hilbert space$${\mathcal{H}}_{\overrightarrow{\lambda }}$$and closely connects to the infinite-dimensional *-representation of the quantum deformed Lorentz group$${\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)$$. The quantum group$${\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)$$also emerges from the quantum gauge transformations of the complex Chern-Simons theory. Focusing on am-holed sphere Σ_0,m, the physical Hilbert space$${\mathcal{H}}_{\text{phys}}$$is identified by imposing the gauge invariance and the flatness constraint. The states in$${\mathcal{H}}_{\text{phys}}$$are the$${\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)$$-invariant linear functionals on a dense domain in$${\mathcal{H}}_{\overrightarrow{\lambda }}$$. Finally, we demonstrate that the physical Hilbert space carries a Fenchel-Nielsen representation, where a set of Wilson loop operators associated with a pants decomposition of Σ_0,mare diagonalized. 

A family of infinite-dimensional irreducible ∗-representations on H≃L2(R)⊗CN is defined for a quantum-deformed Lorentz algebra Uq(sl2)⊗Uq̃(sl2), where q=exp[πiN(1+b2)] and q̃=exp[πiN(1+b−2)] with N∈Z+ and |b| = 1. The representations are constructed with the irreducible representation of quantum torus algebra at level-N, 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) Chern–Simons theory on 4-holed sphere. Finally, we explicitly compute the spectral decomposition of the complex Fenchel–Nielson length operators and the corresponding direct-integral representation of the Hilbert space H, which we call the Fenchel–Nielson representation.