Cell Systems for $\overline{\operatorname{Rep}(U_q(\mathfrak{sl}_N))}$ Module Categories

TL;DR

This paper introduces KW cell systems on graphs to classify module categories over quantum group representations, solving systems for specific cases and constructing new exceptional modules, advancing understanding of quantum subgroups.

Contribution

It defines KW cell systems for classifying module categories over quantum group representations, generalizing previous classification data and solving for complex cases like $sl_4$.

Findings

Classified module categories over $ar{ ext{Rep}}(U_q(sl_4))$

Constructed new exceptional module categories for $sl_4$

Proved the graph planar algebra embedding theorem for oriented planar algebras

Abstract

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.