16a. Small index
We have seen so far the foundations of Jones' subfactor theory, along with results regarding the most basic classes of such subfactors, namely those coming from compact groups, discrete group duals, and more generally compact quantum groups. These subfactors all have integer index, [math]N\in\mathbb N[/math], and appear as subfactors of the Murray-von Neumann hyperfinite [math]{\rm II}_1[/math] factor [math]R[/math], either by definition, or by theorem, or by conjecture.
This suggests looking into the classification of subfactors of integer index, or into the classification of the subfactors of [math]R[/math], or into the classification of the subfactors of [math]R[/math] having integer index. These are all good questions, that we will discuss here.
Before starting, however, and in order to have an idea on what we want to do, we should discuss the following question: should the index [math]N\in[1,\infty)[/math] be small, or big? This is something quite philosophical, and non-trivial, the situation being as follows:
- Mathematics and basic common sense suggest that subfactors should fall into two main classes, “series” and “exceptional”. From this perspective, the series, corresponding to uniform values of the index, must be investigated first.
- In practice now, passed a few simple cases, such as the FC or TL subfactors, we cannot hope for the index to take full uniform values. The more reasonable question here is that of looking at the case where [math]N\in\mathbb N[/math] is uniform.
- The problem now is that, in the lack of theory here, this basically brings us back to groups, group duals, and more generally compact quantum groups, whose combinatorics is notoriously simpler than that of the arbitrary subfactors.
- In short, naivity and pure mathematics tell us to investigate the “big index” case first, but with the remark however that we are missing something, and so that we must do in parallel some study in the “small index” case too.
All this does not look very clear, and so after this discussion, we are basically still in the dark. So, should the answer come then from physics, and applications?
Unfortunately, things here are quite complicated too, basically due to our current poor understanding of quantum mechanics, and of what precisely is to be done, in order to have things in physics moving. And in fact, things here are in fact split too, a bit in the same way as above, the situation being basically as follows:
- The very small index range, [math]N\in[1,4][/math], is subject to the remarkable “quantization” result of Jones, stating that we should have [math]N=4\cos^2(\frac{\pi}{n})[/math], and has strong ties with a number of considerations in conformal field theory.
- In what concerns the other end, [math]N \gt \gt 0[/math], this is in relation with statistical mechanics, once again following work of Jones on the subject, and with the index itself corresponding to physicists' famous “big [math]N[/math]” variable.
In short, no hope for an answer here. At least with our current knowledge of the subject. Probably most illustrating here is the fact that the main experts, starting with Jones himself, have always being split, working on both small and big index.
Getting away now from these philosophical difficulties, and back to our present book, which is rather elementary and mathematical, in this final chapter we will survey the main structure and classification results available, both in small and big index.
As already mentioned, we will focus on the subfactors of the Murray-von Neumann hyperfinite [math]{\rm II}_1[/math] factor [math]R[/math], by taking for granted the fact that these subfactors are the most “important”, and related to physics. With the side remark, however, that this is actually subject to debate too, with many mathematicians opting for bigger factors like [math]L(F_\infty)[/math], and with some physicists joining them too. But let us not get into this here.
In order to get started now, in order to talk about classification, we need invariants for our subfactors. Which brings us into a third controversy, namely the choice between algebraic and analytic invariants. The situation here is as follows:
Associated to any finite index subfactor [math]A\subset B[/math], having planar algebra [math]P=(P_k)[/math], are the following invariants:
- Its principal graph [math]X[/math], which describes the inclusions [math]P_0\subset P_1\subset P_2\subset\ldots\,[/math], with the reflections coming from basic constructions removed.
- Its fusion algebra [math]F[/math], which describes the fusion rules for the various types of bimodules that can appear, namely [math]A-A[/math], [math]A-B[/math], [math]B-A[/math], [math]B-B[/math].
- Its Poincaré series [math]f[/math], which is the generating series of the graded components of the planar algebra, [math]f(z)=\sum_k\dim(P_k)z^k[/math].
- Its spectral measure [math]\mu[/math], which is the probability measure having as moments the dimensions of the planar algebra components, [math]\int x^kd\mu(x)=\dim(P_k)[/math].
This definition is of course something a bit informal, and there is certainly some work to be done, in order to fully define all the above invariants [math]X,F,f,\mu[/math], and to work out the precise relation between them. We will be back to this later, but for the moment, let us keep in mind the fact that associated to a given subfactor [math]A\subset B[/math] are several combinatorial invariants, which are not exactly equivalent, but are definitely versions of the same thing, the “combinatorics of the subfactor”, and which come in algebraic or analytic flavors.
So, what to use? As before, in relation with the previous controversies, the main experts, starting with Jones himself, have always being split themselves on this question, working with both algebraic and analytic invariants. Generally speaking, the algebraic invariants, which are (1) and (2) in the above list, tend to be more popular in small index, while the analytic invariants, (3) and (4), are definitely more popular in big index.
In order to get started now, let us first discuss the question of classifying the subfactors of the hyperfinite [math]{\rm II}_1[/math] factor [math]R[/math], up to isomorphism, having index [math]N\leq4[/math].
This is something quite tricky, and the main idea here will be the fact, coming from the proof of the Jones index restriction theorem, explained in chapter 13 above, that the index [math]N\in(1,4][/math] must be the squared norm of a certain graph:
Now with this observation in hand, the assumption [math]N\leq4[/math] forces [math]X[/math] to be one of the Coxeter-Dynkin graphs of type ADE, and then a lot of work, both of classification and exclusion, leads to an ADE classification for the subfactors of [math]R[/math] having index [math]N\leq 4[/math].
This was for the idea. More in detail now, let us begin by explaining in detail how our subfactor invariant here, which will be the principal graph [math]X[/math], is constructed.
Consider first an arbitrary finite index irreducible subfactor [math]A_0\subset A_1[/math], with associated planar algebra [math]P_k=A_0'\cap A_k[/math], and let us look at the following system of inclusions:
By taking the Bratelli diagram of this system of inclusions, and then deleting the reflections coming from basic constructions, which automatically appear at each step, according to the various results from chapter 13, we obtain a certain graph [math]X[/math], called principal graph of [math]A_0\subset A_1[/math]. The main properties of [math]X[/math] can be summarized as follows:
The principal graph [math]X[/math] has the following properties:
- The higher relative commutant [math]P_k=A_0'\cap A_k[/math] is isomorphic to the abstract vector space spanned by the [math]2k[/math]-loops on [math]X[/math] based at the root.
- In the amenable case, where [math]A_1=R[/math] and when the subfactor is “amenable”, the index of [math]A_0\subset A_1[/math] is given by [math]N=||X||^2[/math].
This is something standard, the idea being as follows:
(1) The statement here, which explains among others the relation between the principal graph [math]X[/math], and the other subfactor invariants, from Definition 16.1 above, comes from the definition of the principal graph, as a Bratelli diagram, with the reflections removed.
(2) This is actually a quite subtle statement, but for our purposes here, we can take the equality [math]N=||X||^2[/math], which reminds a bit the Kesten amenability condition for discrete groups, as a definition for the amenability of the subfactor. With the remark that for the Popa diagonal subfactors what we have here is precisely the Kesten amenability condition for the underlying discrete group [math]\Gamma[/math], and that, more generally, for the arbitrary generalized Popa or Wassermann subfactors, what we have here is precisely the Kesten type amenability condition for the underlying discrete quantum group [math]\Gamma[/math].
As a consequence of the above, in relation with classification questions, we have:
The principal graph of a subfactor having index [math]N\leq 4[/math] must be one of the Coxeter-Dynkin graphs of type ADE.
This follows indeed from the formula [math]N=||X||^2[/math] from the above result, and from the considerations from the proof of the Jones index restriction theorem, explained in chapter 13 above. For full details on all this, we refer for instance to [1].
More in detail now, the usual Coxeter-Dynkin graphs are as follows:
\vskip-7mm
\vskip-7mm
\vskip-9mm
\vskip-7mm Here the graphs [math]A_n[/math] with [math]n\geq 2[/math] and [math]D_n[/math] with [math]n\geq 3[/math] have by definition [math]n[/math] vertices each, [math]\tilde{A}_{2n}[/math] with [math]n\geq 1[/math] has [math]2n[/math] vertices, and [math]\tilde{D}_n[/math] with [math]n\geq 4[/math] has [math]n+1[/math] vertices. Thus, the first graph in each series is by definition as follows:
\vskip-7mm There are also a number of exceptional Coxeter-Dynkin graphs. First we have:
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\vskip-5mm Also, we have as well index 4 versions of the above exceptional graphs, as follows:
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\vskip-5mm Getting back now to Theorem 16.3, with this list in hand, the story is not over here, because we still have to understand which of these graphs can really appear as principal graphs of subfactors. And, for those graphs which can appear, we must understand the structure and classification of the subfactors of [math]R[/math], having them as principal graphs.
In short, there is still a lot of work to be done, as a continuation of Theorem 16.3. The subfactors of index [math]\leq 4[/math] were intensively studied in the 80s and early 90s, and about 10 years after Jones' foundational paper [2], a complete classification result was found, with contributions by many authors. A simplified form of this result is as follows:
The principal graphs of subfactors of index [math]\leq 4[/math] are:
- Index [math] \lt 4[/math] graphs: [math]A_n[/math], [math]D_{even}[/math], [math]E_6[/math], [math]E_8[/math].
- Index [math]4[/math] finite graphs: [math]\tilde{A}_{2n}[/math], [math]\tilde{D}_n[/math], [math]\tilde{E}_6[/math], [math]\tilde{E}_7[/math], [math]\tilde{E}_8[/math].
- Index [math]4[/math] infinite graphs: [math]A_\infty[/math], [math]A_{-\infty,\infty}[/math], [math]D_\infty[/math].
As already mentioned, this is something quite heavy, with contributions by many authors, and among the main papers to be read here, let us mention [2], [3], [4], [5]. Observe that the graphs [math]D_{odd}[/math] and [math]E_7[/math] don't appear in the above list. This is one of the subtle points of subfactor theory. For a discussion here, see [6].
There are many other things that can be said about the subfactors of index [math]N\leq4[/math], both at the theoretical level, of the finite depth and more generally of the amenable subfactors, and at the level of the ADE classification, which makes connections with other ADE classifications. We refer here to [6], [1], [3], [4], [5], [7].
Regarding now the subfactors of index [math]N\in(4,5][/math], and also of small index above 5, these can be classified, but this is a long and complicated story. Let us just record here the result in index 5, which is something quite easy to formulate, as follows:
The principal graphs of the irreducible index [math]5[/math] subfactors are:
- [math]A_\infty[/math], and a non-extremal perturbation of [math]A_\infty^{(1)}[/math].
- The McKay graphs of [math]\mathbb Z_5,D_5,GA_1(5),A_5,S_5[/math].
- The twists of the McKay graphs of [math]A_5,S_5[/math].
As a comment here, the above [math]N=5[/math] result was much harder to obtain than the classification in index [math]N=4[/math], obtained as a consequence of Theorem 16.4. However, at the level of the explicit construction of such subfactors, things are quite similar at [math]N=4[/math] and [math]N=5[/math], with the fixed point subfactors associated to quantum permutation groups [math]G\subset S_N^+[/math] providing most of the examples. We refer here to [9] and related papers.
In index [math]N=6[/math] now, the subfactors cannot be classified, at least in general, due to several uncountable families, coming from groups, group duals, and more generally compact quantum groups. The exact assumption to be added is not known yet.
Summarizing, the current small index classification problem meets considerable difficulties in index [math]N=6[/math], and right below. In small index [math]N \gt 6[/math] the situation is largely unexplored. We refer here to [10] and the recent literature on the subject.
General references
Banica, Teo (2024). "Principles of operator algebras". arXiv:2208.03600 [math.OA].
References
- 1.0 1.1 F.M. Goodman, P. de la Harpe and V.F.R. Jones, Coxeter graphs and towers of algebras, Springer (1989).
- 2.0 2.1 V.F.R. Jones, Index for subfactors, Invent. Math. 72 (1983), 1--25.
- 3.0 3.1 A. Ocneanu, Quantized groups, string algebras and Galois theory for algebras, London Math. Soc. Lect. Notes 136 (1988), 119--172.
- 4.0 4.1 A. Ocneanu, Quantum symmetry, differential geometry of finite graphs, and classification of subfactors, Univ. Tokyo Seminar Notes (1990).
- 5.0 5.1 S. Popa, Classification of subfactors: the reduction to commuting squares, Invent. Math. 101 (1990), 19--43.
- 6.0 6.1 D.E. Evans and Y. Kawahigashi, Quantum symmetries on operator algebras, Oxford Univ. Press (1998).
- S. Popa, Classification of amenable subfactors of type II, Acta Math. 172 (1994), 163--255.
- 8.0 8.1 V.F.R. Jones, S. Morrison and N. Snyder, The classification of subfactors of index at most 5, Bull. Amer. Math. Soc. 51 (2014), 277--327.
- T. Banica, J. Bichon and B. Collins, The hyperoctahedral quantum group, J. Ramanujan Math. Soc. 22 (2007), 345--384.
- Z. Liu, S. Morrison and D. Penneys, 1-supertransitive subfactors with index at most [math]6\frac{1}{5}[/math], Comm. Math. Phys. 334 (2015), 889--922.