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Singly generated planar algebras of small dimension. (English) Zbl 1075.46053
Summary: Planar algebras have been introduced by the second author as a tool to study subfactors. A planar algebra encodes the standard invariant of a subfactor in such a way that combinatorial and algebraic studies are handled in a graphical method.
The authors study simplest subfactors from the viewpoint of planar algebras. That is, they study subfactors \(N\subset M\) such that the corresponding planar algebra is generated by a single element in \(N'\cap M_1\), \(\dim (N'\cap M_1)=3\), and \(\dim (N'\cap M_2)\leq 12\). In the course of this study, they obtain a certain sufficient condition for existence of a nontrivial intermediate subfactor. Using this, they reach the main classification result which says that if a planar algebra satisfies the above three conditions then it is a Fuss-Catalan algebra studied previously by the authors in [Invent. Math. 128, No. 1, 89–157 (1997; Zbl 0891.46035)], except for the easy case of a subfactor of index 3. Several graphical techniques are introduced for this purpose.
[For part II see Adv. Math. 175, No. 2, 297–318 (2003; Zbl 1041.46048).]

MSC:
46L37 Subfactors and their classification
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