2018/10/16 by Desmond Coles, Peter Huston, Coles, Desmond +5 · 1 citation
Mathematics · #18D50 (Secondary) #46L37 (Primary) 18D05 #Category Theory (math.CT) #FOS: Mathematics #Operator Algebras (math.OA) #Quantum Algebra (math.QA) #math.CT #math.OA #math.QA #msc:18D05 #msc:18D50 #msc:46L37
paper · pdf · doi:10.48550/arxiv.1810.07049
40 pages, many figures, comments welcome
arxiv created 2018/10/16 · arxiv updated 2018/10/17
Jones and Penneys showed that a finite depth subfactor planar algebra embeds in the bipartite graph planar algebra of its principal graph, via a Markov towers of algebras approach. We relate several equivalent perspectives on the notion of module over a subfactor planar algebra, and show that a Markov tower is equivalent to a module over the Temperley-Lieb-Jones planar algebra. As a corollary, we obtain a classification of semisimple pivotal C* modules over Temperley-Lieb-Jones in terms of pointed graphs with a Frobenius-Perron vertex weighting. We then generalize the Markov towers of algebras approach to show that a finite depth subfactor planar algebra embeds in the bipartite graph planar algebra of the fusion graph of any of its cyclic modules.