The Planar Algebra of a bipartite graph

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Cohomology and $L^2$-Betti numbers for subfactors and quasi-regular inclusions (2015, background): “tly the higher relative commutants T′ ∩ Sk associated to the Jones tower T ⊂ S ⊂ S1 ⊂ ···. The diﬀerential of this complex is then precisely the one considered in [J98, Section 6] and it follows from [J98] that the complex is acyclic. Remark 9.12. Let T ⊂ S be an irreducible quasi-regular inclusion ”
Quantum automorphism groups of small metric spaces (2003, background): “nar algebra satisfying the following conditions. (1′) dim(P 1) = 1 (“transitivity”) and P is generated by a 2-box b ∈ P 2 (“universality”). (2′) P embeds into the depth 1 planar algebra associated in [9] to the inclusion C ⊂ C(X) and b corresponds in this way to a distance matrix (“tannakian conditi”