Quantum Teichmüller spaces and Kashaev's $6j$–symbols

Abstract

The Kashaev invariants of 3–manifolds are based on $6j$–symbols from the representation theory of the Weyl algebra, a Hopf algebra corresponding to the Borel subalgebra of $U_q\left(sl\left(2,ℂ\right)\right)$. In this paper, we show that Kashaev’s $6j$–symbols are intertwining operators of local representations of quantum Teichmüller spaces. This relates Kashaev’s work with the theory of quantum Teichmüller space, which was developed by Chekhov–Fock, Kashaev and continued by Bonahon–Liu.