The construction of a E 7-like quantum subgroup of SU(3)

Abstract

In this short note we construct an embedding of the planar algebra for Rep ( U q ( s l 3 ) ) ¯ at q = e 2 π i 1 24 into the graph planar algebra of di Francesco and Zuber’s candidate graph E 4 12 . Via the graph planar algebra embedding theorem we thus construct a rank 11 module category over Rep ( U q ( s l 3 ) ) ¯ whose graph for action by the vector representation is E 4 12 . This fills a small gap in the literature on the construction of Rep ( U q ( s l 3 ) ) ¯ module categories. As a consequence of our construction, we obtain the principal graphs of subfactors constructed abstractly by Evans and Pugh.