Finite dimensional quantum Teichm\"uller space from the quantum torus at root of unity

Representation theory of the quantum torus Hopf algebra, when the parameter $q$ is a root of unity, is studied. We investigate a decomposition map of the tensor product of two irreducibles into the direct sum of irreducibles, realized as a `multiplicity module' tensored with an irreducible representation. The isomorphism between the two possible decompositions of the triple tensor product yields a map ${\bf T}$ between the multiplicity modules, called the 6j-symbols. We study the left and right dual representations, and correspondingly, the left and right representations on the ${\rm Hom}$ spaces of linear maps between representations. Using the isomorphisms of irreducibles to left and right duals, we construct a map ${\bf A}$ on a multiplicity module, encoding the permutation of the roles of the irreducible representations in the identification of the multiplicity module as the space of intertwiners between representations. We show that ${\bf T}$ and ${\bf A}$ satisfy certain consistency relations, forming a Kashaev-type quantization of the Teichm\"uller spaces of bordered Riemann surfaces. All constructions and proofs in the present work uses only plain representation theoretic language with the help of the notions of the left and the right dual and Hom representations, and therefore can be applied easily to other Hopf algebras for future works.