Standard -lattices, rigid tensor categories, and (bi)modules

  • Quan Chen

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

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Quan Chen, Standard -lattices, rigid tensor categories, and (bi)modules. DM 29 (2024), no. 2, pp. 247–341

DOI 10.4171/DM/944