  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.
