Cell systems for $\overline{\operatorname{Rep}(U_q(\mathfrak{sl}_N))}$ module categories

In this paper, we define the KW cell system on a graph \Gamma , depending on parameters N\in \mathbb{N} , q a root of unity, and \omega an N -th root of unity. This is a polynomial system of equations depending on \Gamma and the parameters. Using the graph planar algebra embedding theorem, we prove that when q = e^{2\pi i \frac{1}{2(N+k)}} , solutions to the KW cell system on \Gamma classify module categories over \overline{\operatorname{Rep}(U_q(\mathfrak{sl}_N))^\omega} whose fusion graph for the object \Lambda_{1} is \Gamma . The KW cell system is a generalisation of the Etingof–Ostrik and the De Commer–Yamashita classifying data for \overline{\operatorname{Rep}(U_q(\mathfrak{sl}_2))} module categories, and Ocneanu’s cell calculus for \overline{\operatorname{Rep}(U_q(\mathfrak{sl}_3))} module categories. To demonstrate the effectiveness of this cell calculus, we solve the KW cell systems corresponding to the exceptional module categories over \overline{\operatorname{Rep}(U_q(\mathfrak{sl}_4))} when q= e^{2\pi i \frac{1}{2(4+k)}} , as well as for all three infinite families of charge-conjugation modules. Building on the work of the second author, this explicitly constructs and classifies all irreducible module categories over \mathcal{C}(\mathfrak{sl}_{4}, k) for all k\in \mathbb{N} . These results prove claims made by Ocneanu on the quantum subgroups of \mathit{SU}(4) . We also construct exceptional module categories over \overline{\operatorname{Rep}(U_q(\mathfrak{sl}_4))^\omega} where \omega\in \{-1, \mathbf{i}, -\mathbf{i}\} . Two of these module categories have no analogue when \omega=1 . The main technical contributions of this paper are a proof of the graph planar algebra embedding theorem for oriented planar algebras, and a refinement of Kazhdan and Wenzl’s skein theory presentation of the category \overline{\operatorname{Rep}(U_q(\mathfrak{sl}_N))^\omega} . We also explicitly describe the subfactors coming from a solution to a KW cell system.