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Planar algebras. I. (English) Zbl 1328.46049
A subfactor \(N \subset M\) is an inclusion of two von Neumann algebras with trivial centers. The first examples of subfactors come from taking \(N\) to be the fixed points for group \(G\) acting on a factor \(M\). In this situation, \(G\) is finite if and only if \(M\) is finitely generated as an \(N\)-module. Furthermore, one can recover the size of \(G\) as the von Neumann dimension of \(M\) as a left \(N\)-module. V. F. R. Jones defined a notion of index for subfactors [Invent. Math. 72, 1–25 (1983; Zbl 0508.46040)], where we say that \(N \subset M\) is finite index if \(M\) is finitely generated, and we define the index to be the von Neumann dimension of \(M\) as an \(N\)-module. This index is a positive real number which may not be an integer, and subfactors of non-integer index cannot come from groups. The first big surprise in the field was Jones’s index theorem, which states that the index cannot be an arbitrary real number, but instead must be of the form \(4 \cos^2 \left(\frac{\pi}{k}\right)\) or lie in \([4, \infty)\). Furthermore, the indices below \(4\) are realized by subfactors constructed out of the Temperley-Lieb-Jones algebras (henceforth, TLJ algebras). These algebras in turn are closely related to quantum groups, and so one can think of subfactors as coming from fixed points of quantum group actions.
Following the above ideas, the study of subfactors can be broken up into two steps: finding algebraic gadgets playing the role of a group and understanding the ways these gadgets be realized as acting on a particular factor. The algebraic gadget attached to a subfactor is called the standard invariant. This standard invariant consists of all maps between all \(N\)-\(N\), \(N\)-\(M\), \(M\)-\(N\), and \(M\)-\(M\) bimodules which appear as summands of tensor powers \(M \otimes_N M \otimes_N\dots \otimes_N M\). These maps have a lot of algebraic structure, and the standard invariant has been axiomatized in several ways including A. Ocneanu’s paragroups [Lond. Math. Soc. Lect. Note Ser. 136, 119–172 (1988; Zbl 0696.46048)] and S. Popa’s standard \(\lambda\)-lattices [Invent. Math. 120, No. 3, 427–445 (1995; Zbl 0831.46069)]. The main goal of the paper under review is to introduce a new axiomatization for the standard invariant of a subfactor called “subfactor planar algebras.” In this approach, the standard invariant of a subfactor consists of certain diagrams drawn in the plane which can be “multiplied” in an enormous number of ways by connecting up their boundaries in a non-crossing fashion. This plethora of multiplications can be made precise by using the notion of a colored operad [J. P. May, Contemp. Math. 202, 1–7 (1997; Zbl 0879.18002)].
The planar algebraic description of standard invariants is heavily inspired by knot theory. The Jones polynomial invariant of links was originally constructed from the TLJ algebras purely algebraically by seeing that the relations that the algebra generators satisfy closely resemble the braid relations [V. F. R. Jones, Bull. Am. Math. Soc., New Ser. 12, 103–111 (1985; Zbl 0564.57006)]. Kauffman observed that the TLJ algebras can be described topologically, and that this description simplifies the construction of the Jones polynomial leading to the L. H. Kauffman bracket description [Topology 26, 395–407 (1987; Zbl 0622.57004)]. Explicitly, the \(n\)th TLJ algebra with index \(d^2\) consists of formal linear combinations of non-crossing tangles with \(n\) boundary points at the bottom and \(n\) boundary points at the top, with multiplication given by gluing the top boundary of one diagram to the bottom boundary of the other diagram and using the relation that you can remove a closed loop and multiply by the number \(d\). The planar algebraic structure comes from the fact that non-crossing tangles can be glued together in many more ways by connecting up the boundary points in any planar way. Thus the planar algebraic approach to subfactors has much in common with knot polynomials, skein theory, and quantum topology. In particular, Jones’s axiomatization is closely related to G. Kuperberg’s spiders [Commun. Math. Phys. 180, No. 1, 109–151 (1996; Zbl 0870.17005)].
Planar algebras can be defined in a great deal of generality (indeed, they can be thought of as an axiomatization of \(2\)-categories with duality), but this paper concentrates on subfactor planar algebras which have some extra structure corresponding specifically to the subfactor case. For example, because we want to consider \(N\)-\(N\), \(N\)-\(M\), \(M\)-\(N\), and \(M\)-\(M\) bimodules, every diagram has a shading of the regions into \(N\)-regions and \(M\)-regions. Similarly, since \(N\) and \(M\) have trivial centers, we have an identification of the space of \(0\)-boundary point pictures with the complex numbers. Finally, since all the spaces of maps are Hilbert spaces, we have an involution and a positive definite pairing on all the spaces.
The paper under review begins with a long introduction which motivates subfactor planar algebras and sketches their definition. Section 1 gives precise definitions of the key notions of the planar operad, general planar algebras, and subfactor planar algebras. Section 2 gives a large number of rich examples of planar algebras. These planar algebras come from many different sources: subfactors, topology, combinatorial group theory, and statistical mechanics. These examples illustrate the fundamental importance of planar algebras throughout mathematics. This section includes a long discussion of biunitary connections and how they relate to planar algebras. Section 3 develops some structural tools used to study subfactor planar algebras, including the usual ring structure, the Markov trace, principal graphs, rotations, free products, cabling, and the tensor product. The main result of Section 4 is that the standard invariant of a subfactor has the structure of a subfactor planar algebra. Section 4 also includes a discussion of the action of annular tangles on planar algebras (which was further developed in [V. F. R. Jones, Monogr. Enseign. Math. 38, 401–463 (2001; Zbl 1019.46036)]) and a sketch of why any subfactor planar algebra comes from at least one subfactor by a short reduction to a deep result of Popa [loc. cit.].
The theory of subfactor planar algebras introduced in this paper has had a large impact on the field, which I will briefly survey. The first major impact is that it suggests studying subfactors in the spirit of J. H. Conway’s skein theory approach to knot theory [Proc. Conf. Oxford 1967, 329–358 (1970; Zbl 0202.54703)]. That is, one starts with a collection of generating tangles (in knot theory, the generator is the crossing) and then imposes certain skein relations (in knot theory, these are the Reidemeister moves together with the additional skein relations satisfied by a specific knot polynomial). One of the first major successes in this direction was the Bisch-Jones skein theoretic description of the standard invariant of a subfactor which has an intermediate subfactor \(N \subset P \subset M\). Bisch had shown that the projection onto \(P\) satisfies an additional natural condition, and Bisch-Jones showed that this condition has a very natural skein theoretic interpretation where the generator is the projection \(P\) thought of as a 4-boundary point diagram. This allowed them to construct a Fuss-Catalan planar algebra which plays the same role for intermediate subfactors that the TLJ planar algebra plays for ordinary subfactors. Continuing in this direction, Bisch, Jones, and later Liu [D. Bisch and V. Jones, Duke Math. J. 101, No. 1, 41–75 (2000; Zbl 1075.46053); Adv. Math. 175, No. 2, 297–318 (2003; Zbl 1041.46048)]; [D. Bisch, Z.-W. Liu and V. Jones, “Singly generated planar algebras of small dimension, Part III,” arXiv:1410.2876]; [Z.-W. Liu, “Singly generated planar algebras of small dimension, Part IV,” arXiv:1507.06030] classified all “small” planar algebras generated by a \(4\)-box. In addition to the Fuss-Catalan planar algebra, there are also planar algebras related to the BMW algebras, and some new planar algebras related to conformal inclusions. These skein theoretic approaches have made the field of subfactors more accessible to topologists which has led to a rich interplay of ideas between the two fields. This interplay has gone both directions, for example, planar algebras played a role in the development of Khovanov homology due to their appearance in [D. Bar-Natan, Geom. Topol. 9, 1443–1499 (2005; Zbl 1084.57011)].
A second major theme in subsequent work in planar algebras is that previous approaches to standard invariants had emphasized just one of the many planar “multiplications” at the expense of the importance of the other multiplications. In particular, the planar algebraic approach has illuminated the key role played by rotation operators and annular Temperley-Lieb diagrams. For example, the rotational point of view led to improved triple point obstructions in [V. F. R. Jones, Duke Math. J. 161, No. 12, 2257–2295 (2012; Zbl 1257.46033)], [N. Snyder, Anal. PDE 6, No. 8, 1923–1928 (2013; Zbl 1297.46044)], [D. Penneys, Adv. Math. 273, 32–55 (2015; Zbl 1326.46049)]. A second important new “multiplication” is a graded tensor product, which has played a key role in finding a connection between free probability and planar algebras in random matrices, free probability, planar algebras and subfactors (cf. [A. Guionnet et al., Clay Math. Proc. 11, 201–239 (2010; Zbl 1219.46057)] and subsequent papers).
Planar algebras have played a key role in recent progress in the classification of small index subfactors. Above index \(4\), there are two families of subfactor planar algebras that exist for every such index, but U. Haagerup [in: Subfactors. Proceedings of the Taniguchi symposium on operator algebras, Kyuzeso, Japan, July 6–10, 1993. Singapore: World Scientific. 1–38 (1994; Zbl 0933.46058)] realized that once you exclude these two families, the possible indices are again quantized between index \(4\) and \(3+\sqrt{3}\). Building on the ideas in the paper under review and the subsequent developments mentioned above, this work has been extended to give a complete classification of subfactor planar algebras up to index \(5.25\) (see [V. F. R. Jones, Bull. Am. Math. Soc., New Ser. 51, No. 2, 277–327 (2014; Zbl 1301.46039)] for a survey of the classification up to index 5, and upcoming work of Afzaly-Morrison-Penneys for index \(5.25\)). Following Haagerup, the main technique is to list the possible principal graphs which pass certain basic tests, then exclude some of these candidates by using more sophisticated tests, and finally construct standard invariants in the few remaining cases. Planar algebras played a key role in this project in three ways. First, triple point obstructions are one of the two main tests used in the initial enumeration, and the improved triple point obstructions using the rotational planar algebraic viewpoint are needed. Second, at index exactly \(3+\sqrt{5}\) things behave quite differently because this number is a product of two allowed indices (this also happens at index \(4\) and index \(6\), yielding many subfactors related to dihedral groups and quotients of \(\mathrm{PSL}_2(\mathbb{Z})\), respectively). There was a family of potential planar algebras at index \(3+\sqrt{5}\) found by Bisch-Haagerup, and all but the three smallest candidates in this family were ruled out by Liu in a tour-de-force planar algebraic calculation [Z.-W. Liu, Adv. Math. 279, 307–371 (2015; Zbl 1330.46061)]. Finally, planar algebras played a key role in the construction of some of the exceptional examples which were discovered through the classification project. These constructions (following an outline of Jones and Peters) first study the planar algebra by breaking it up as representations of the category of annular Temperley-Lieb diagrams ([V. F. R. Jones, Duke Math. J. 161, No. 12, 2257–2295 (2012; Zbl 1257.46033)], [E. Peters, Int. J. Math. 21, No. 8, 987–1045 (2010; Zbl 1203.46039)]). For the construction of the extended Haagerup subfactor [S. Bigelow et al., Acta Math. 209, No. 1, 29–82 (2012; Zbl 1270.46058)], an additional planar algebraic technique was needed, namely, a new skein theory due to Bigelow called the jellyfish algorithm. In complete contrast to all previously known skein theories, which at each step either decrease the number of generators or keep the number the same and move them into a more favorable configuration, the jellyfish algorithm works by rapidly increasing the number generators in the diagrams but moving these generators closer to the outside of the diagram where they eventually all simplify.

46L37 Subfactors and their classification
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