The Witten-Reshetikhin-Turaev representation of the Kauffman bracket skein algebra (Q2796745)

The paper under review forms a chapter in the research of the authors on the finite dimensional irreducible representations of the Kauffman bracket skein algebra \(\mathcal{S}^A(S)\) of an oriented surface \(S\). Witten-Reshetikhin-Turaev (WRT) topological quantum field theory provides a representation \(\rho: \mathcal{S}^A(S)\rightarrow \mathrm{End}(V_S)\) for every primitive \(2N\)-root of unity \(A\). The paper proves that this representation is irreducible when \(S\) is connected and closed. Roberts previously proved this result for \(N=2p\) with \(p\) prime [\textit{J. Roberts}, J. Knot Theory Ramifications 10, No. 5, 763--767 (2001; Zbl 1001.57036)].NEWLINENEWLINENext, as in previous work of the authors, for \(N\) odd, the authors associate to \(\rho\) an element \(r_\rho\) of the character variety \(\mathcal{R}_{SL_2(\mathbb{C})}(S)\). Considering \(\mathcal{S}^A(S)\) as a quantization of \(\mathcal{R}_{SL_2(\mathbb{C})}(S)\) motivates naming \(r_\rho\) the \textit{classical shadow} of \(\rho\). The authors prove that the classical shadow of the WRT representation is the trivial character represented by the trivial homomorphism \(\pi_1(S)\rightarrow SL_2(\mathbb{C})\).