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 $\overline{\mathrm{Rep}(U_q(sl_N))^\omega}$ whose action 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 $\overline{\mathrm{Rep}(U_q(sl_2))}$ module categories, and Ocneanu's cell calculus for $\overline{\mathrm{Rep}(U_q(sl_3))}$ module categories. To demonstrate the effectiveness of this cell calculus, we solve the KW cell systems corresponding to the exceptional module categories over $\overline{\mathrm{Rep}(U_q(sl_4))}$ 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 $\mathcal{C}(sl_4, k)$ for all $k\in \mathbb{N}$. These results prove claims made by Ocneanu on the quantum subgroups of $SU(4)$. We also construct exceptional module categories over $\overline{\mathrm{Rep}(U_q(sl_4))^\omega}$ where $\omega\in \{-1, i, -i\}$. 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 $\overline{\mathrm{Rep}(U_q(sl_N))^\omega}$. We also explicitly describe the subfactors coming from a solution to a KW cell system.