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arXiv · 2308.16849

The construction of a $E_7$-like quantum subgroup of $SU(3)$

Cain Edie-Michell·Lance Marinelli
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Abstract

In this short note we construct an embedding of the planar algebra for $\overline{\operatorname{Rep}(U_q(sl_3))}$ at $q = e^{2\pi i \frac{1}{24}}$ into the graph planar algebra of di Francesco and Zuber's candidate graph $\mathcal{E}_4^{12}$. Via the graph planar algebra embedding theorem we thus construct a rank 11 module category over $\overline{\operatorname{Rep}(U_q(sl_3))}$ whose graph for action by the vector representation is $\mathcal{E}_4^{12}$. This fills a small gap in the literature on the construction of $\overline{\operatorname{Rep}(U_q(sl_3))}$ module categories. As a consequence of our construction, we obtain the principal graphs of subfactors constructed abstractly by Evans and Pugh.

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Cain Edie-Michell, Lance Marinelli. 2023-08-31. The construction of a $E_7$-like quantum subgroup of $SU(3)$. https://arxiv.org/abs/2308.16849

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