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

Abstract

In this article, we construct a $2$-shaded rigid ${\mathrm{C}}^{\ast }$ multitensor category with canonical unitary dual functor directly from a standard $\lambda$-lattice. We use the notions of traceless Markov towers and lattices to define the notion of module and bimodule over standard $\lambda$-lattice(s), and we explicitly construct the associated module category and bimodule category over the corresponding $2$-shaded rigid ${\mathrm{C}}^{\ast }$ multitensor category. As an example, we compute the modules and bimodules for Temperley–Lieb–Jones standard $\lambda$-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 $\mathcal{T}\mathcal{L}\mathcal{J}$ module categories in terms of edge weighted graphs, and a classification of $\mathcal{T}\mathcal{L}\mathcal{J}$ 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.