Standard $\lambda$-lattices, rigid $\mathrm{C}^{*}$ tensor categories, and (bi)modules

Quan Chen

Documenta mathematica, Tome 29 (2024) no. 2, pp. 247-341

Résumé

In this article, we construct a 2-shaded rigid C∗ multitensor category with canonical unitary dual functor directly from a standard λ-lattice. We use the notions of traceless Markov towers and lattices to define the notion of module and bimodule over standard λ-lattice(s), and we explicitly construct the associated module category and bimodule category over the corresponding 2-shaded rigid C∗ multitensor category. As an example, we compute the modules and bimodules for Temperley–Lieb–Jones standard λ-lattices in terms of traceless Markov towers and lattices. Translating into the unitary 2-category of bigraded Hilbert spaces, we recover De Commer–Yamashita’s classification of TLJ module categories in terms of edge weighted graphs, and a classification of TLJ bimodule categories in terms of biunitary connections on square-partite weighted graphs. As an application, we show that every (infinite depth) subfactor planar algebra embeds into the bipartite graph planar algebra of its principal graph.

Détail

DOI : 10.4171/dm/944

Classification : 46L37, 18M30, 18N10
Mots-clés : Rigid C∗ tensor category, module category, bimodule category, standard λ-lattice, Markov tower, subfactor, planar algebra, module embedding