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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 thetensor product of two irreducibles into the direct sum of irreducibles,realized as a `multiplicity module' tensored with an irreduciblerepresentation. The isomorphism between the two possible decompositions of thetriple tensor product yields a map ${\bf T}$ between the multiplicity modules,called the 6j-symbols. We study the left and right dual representations, andcorrespondingly, the left and right representations on the ${\rm Hom}$ spacesof linear maps between representations. Using the isomorphisms of irreduciblesto 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 theidentification of the multiplicity module as the space of intertwiners betweenrepresentations. We show that ${\bf T}$ and ${\bf A}$ satisfy certainconsistency relations, forming a Kashaev-type quantization of the Teichm\"ullerspaces of bordered Riemann surfaces. All constructions and proofs in thepresent work uses only plain representation theoretic language with the help ofthe notions of the left and the right dual and Hom representations, andtherefore can be applied easily to other Hopf algebras for future works.

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