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The Kashaev invariants of 3-manifolds are based on $6j$-symbols from therepresentation theory of the Weyl algebra, a Hopf algebra corresponding to theBorel subalgebra of $U_q(sl(2,\C))$. In this paper, we show that Kashaev's$6j$-symbols are intertwining operators of local representations of quantumTeichm\"uller spaces. This relates Kashaev's work with the theory of quantumTeichm\"uller space, which was developed by Chekhov-Fock, Kashaev and continuedby Bonahon-Liu.

Quantum Teichm\"uller spaces and Kashaev's 6j-symbols