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In what follows we discuss various structure and classification questions for the subfactors, all interesting questions, related to physics, regarded from a probabilistic viewpoint. In order to get started, we need invariants for our subfactors. We have the choice here between algebraic and analytic invariants, the situation being as follows:

Definition

Associated to any finite index subfactor [math]A_0\subset A_1[/math], having planar algebra [math]P=(P_n)[/math], are the following invariants:

• Its principal graph [math]\Gamma[/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_0-A_0[/math], [math]A_0-A_1[/math], [math]A_1-A_0[/math], [math]A_1-A_1[/math].
• Its Poincaré series [math]f[/math], which is the generating series of the graded components of the planar algebra, [math]f(z)=\sum_n\dim(P_n)z^n[/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^nd\mu(x)=\dim(P_n)[/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 these invariants [math]\Gamma,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 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.

More in detail now, let us begin by explaining how the principal graph [math]\Gamma[/math] is constructed. Consider a finite index irreducible subfactor [math]A_0\subset A_1[/math], with associated planar algebra [math]P_n=A_0'\cap A_n[/math], and let us look at the following system of inclusions:

[[math]] P_0\subset P_1\subset P_2\subset\ldots [[/math]]

By taking the Bratelli diagram of this system of inclusions, and then deleting the reflections coming from basic constructions, we obtain a certain graph [math]\Gamma[/math], called principal graph of [math]A_0\subset A_1[/math]. The main properties of [math]\Gamma[/math] can be summarized as follows:

Proposition

The principal graph [math]\Gamma[/math] has the following properties:

• The higher relative commutant [math]P_n=A_0'\cap A_n[/math] is isomorphic to the abstract vector space spanned by the [math]2n[/math]-loops on [math]\Gamma[/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=||\Gamma||^2[/math].

Show Proof

This is something standard, the idea being as follows:

(1) The statement here, which explains among others the relation between the principal graph [math]\Gamma[/math], and the other subfactor invariants, from Definition 16.36, 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=||\Gamma||^2[/math], which reminds the Kesten amenability condition for discrete groups, as a definition for the amenability of the subfactor.

(3) 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]G[/math].

(4) And with the further remark that, more generally, for the arbitrary generalized Popa or Wassermann subfactors, discussed above, what we have here is precisely the Kesten type amenability condition for the underlying discrete quantum group [math]G[/math].

■

As an illustration for all this, let us first discuss the case of the small index subfactors, [math]N\in[1,4][/math]. Following Jones and related work, we first have the following result:

Theorem

The index of subfactors is subject to the condition

[[math]] N\in\left\{4\cos^2\left(\frac{\pi}{n}\right)\Big|n\geq3\right\}\cup[4,\infty] [[/math]]

and at [math]N\leq4[/math], the principal graph must be one of the Coxeter-Dynkin ADE graphs.

Show Proof

This comes from the combinatorics of [math]e_1,e_2,e_3,\ldots\,[/math], as folows:

(1) In order to best comment on what happens, when iterating the basic construction, let us record the first few values of the numbers in the statement, namely:

[[math]] 4\cos^2\left(\frac{\pi}{3}\right)=1\quad,\quad 4\cos^2\left(\frac{\pi}{4}\right)=2 [[/math]]

[[math]] 4\cos^2\left(\frac{\pi}{5}\right)=\frac{3+\sqrt{5}}{2}\quad,\quad 4\cos^2\left(\frac{\pi}{6}\right)=3 [[/math]]

(2) By using a basic construction, we get, by trace manipulations on [math]e_1[/math]:

[[math]] N\notin(1,2) [[/math]]

With a double basic construction, we get, by trace manipulations on [math] \lt e_1,e_2 \gt [/math]:

[[math]] N\notin\left(2,\frac{3+\sqrt{5}}{2}\right) [[/math]]

And so on. In short, by doing computations, we are led to the conclusion in the statement, by a kind of recurrence, involving a certain family of orthogonal polynomials.

(3) In practice now, following , the most elegant way of proving the result is by using the fact, explained in Theorem 16.14, that that sequence of Jones projections [math]e_1,e_2,e_3,\ldots\subset B(H)[/math] generates a copy of the Temperley-Lieb algebra of index [math]N[/math]:

[[math]] TL_N\subset B(H) [[/math]]

With this result in hand, we must prove that such a representation cannot exist in index [math]N \lt 4[/math], unless we are in the following special situation:

[[math]] N=4\cos^2\left(\frac{\pi}{n}\right) [[/math]]

But this can be proved by using some suitable trace and positivity manipulations on [math]TL_N[/math], as in (2) above, and for full details here, we refer to Jones' paper .

(4) As for the second assertion in the statement, this comes via a refinement of all this, the key ingredient being the fact that in index [math]N\leq4[/math], and in fact more generally in the amenable case, as discussed before, we must have [math]N=||\Gamma||^2[/math]. See .

■

More in detail now, the usual Coxeter-Dynkin ADE graphs are as follows:

[[math]] A_k=\bullet-\circ-\circ\cdots\circ-\circ-\circ\hskip20mm A_{\infty}=\bullet-\circ-\circ-\circ\cdots\hskip7mm [[/math]]

\vskip-7mm

[[math]] D_k=\bullet-\circ-\circ\dots\circ- \begin{matrix}\ \circ\cr\ |\cr\ \circ \cr\ \cr\ \end{matrix}-\circ\hskip70mm [[/math]]

\vskip-7mm

[[math]] \ \ \ \ \ \ \ \tilde{A}_{2k}= \begin{matrix} \circ&\!\!\!\!-\circ-\circ\cdots\circ-\circ-&\!\!\!\!\circ\cr |&&\!\!\!\!|\cr \bullet&\!\!\!\!-\circ-\circ-\circ-\circ-&\!\!\!\!\circ\cr\cr\cr\end{matrix}\hskip15mm A_{-\infty,\infty}= \begin{matrix} \circ&\!\!\!\!-\circ-\circ-\circ\cdots\cr |&\cr \bullet&\!\!\!\!-\circ-\circ-\circ\cdots\cr\cr\cr\end{matrix} \hskip15mm [[/math]]

\vskip-9mm

[[math]] \;\tilde{D}_k=\bullet- \begin{matrix}\circ\cr|\cr\circ\cr\ \cr\ \end{matrix}-\circ\dots\circ- \begin{matrix}\ \circ\cr\ |\cr\ \circ \cr\ \cr\ \end{matrix}-\circ \hskip20mm D_\infty=\bullet- \begin{matrix}\circ\cr|\cr\circ\cr\ \cr\ \end{matrix}-\circ-\circ\cdots\hskip7mm [[/math]]

\vskip-7mm There are as well a number of exceptional Coxeter-Dynkin graphs. First we have:

[[math]] E_6=\bullet-\circ- \begin{matrix}\circ\cr|\cr\circ\cr\ \cr\ \end{matrix}- \circ-\circ\hskip71mm [[/math]]

\vskip-13mm

[[math]] E_7=\bullet-\circ-\circ- \begin{matrix}\circ\cr|\cr\circ\cr\ \cr\ \end{matrix}- \circ-\circ\hskip18mm [[/math]]

\vskip-15mm

[[math]] \hskip30mm E_8=\bullet-\circ-\circ-\circ- \begin{matrix}\circ\cr|\cr\circ\cr\ \cr\ \end{matrix}- \circ-\circ [[/math]]

\vskip-5mm Finally, we have index 4 versions of the above exceptional graphs, as follows:

[[math]] \tilde{E}_6=\bullet-\circ-\begin{matrix} \circ\cr| \cr\circ\cr|&\cr\circ&\!\!\!\!-\ \circ\cr\ \cr\ \cr\ \cr\ \end{matrix}-\circ\hskip71mm [[/math]]

\vskip-22mm

[[math]] \tilde{E}_7=\bullet-\circ-\circ- \begin{matrix}\circ\cr|\cr\circ\cr\ \cr\ \end{matrix}- \circ-\circ-\circ\hskip18mm [[/math]]

\vskip-15mm

[[math]] \hskip30mm \tilde{E}_8=\bullet-\circ-\circ-\circ-\circ- \begin{matrix}\circ\cr|\cr\circ\cr\ \cr\ \end{matrix}- \circ-\circ [[/math]]

\vskip-5mm Getting back now to Theorem 16.38, with this list in hand, the story is not over, 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, still a lot of work to be done, as a continuation of Theorem 16.38. 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 , a complete classification result was found, with contributions by many authors. A simplified form of this result is as follows:

Theorem

The principal graphs of subfactors of index [math]\leq 4[/math] are:

• Index [math] \lt 4[/math] graphs: [math]A_k[/math], [math]D_{even}[/math], [math]E_6[/math], [math]E_8[/math].
• Index [math]4[/math] finite graphs: [math]\tilde{A}_{2k}[/math], [math]\tilde{D}_k[/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].

Show Proof

As already mentioned, this is something quite heavy, with contributions by many authors, and notably Ocneanu . Observe that, as a subtlety of subfactor theory, the graphs [math]D_{odd}[/math] and [math]E_7[/math] don't appear in the above list. For a discussion, see .

■

With the above understood, we can now have a more conceptual look at the random walk computations from chapter 3. Let us recall indeed from there that we have:

Definition

The Poincaré series of a rooted bipartite graph [math]X[/math] is

[[math]] f(z)=\sum_{k=0}^\infty L_{2k}z^k [[/math]]

where [math]L_{2k}[/math] is the number of [math]2k[/math]-loops based at the root.

We can see that this is in tune with Definition 16.36, in the sense that the Poincaré series constructed there coincides with the above one, with [math]X[/math] being the principal graph. Thus, when looking now at the spectral measures, these coincide too, and we have: \begin{conclusion} The spectral measures of ADE graphs that we computed in chapter 3 are, from a subfactor viewpoint, the spectral measures of subfactors of index [math]\leq4[/math]. \end{conclusion} Which is certainly something very nice, and for the continuation of the story here, we refer to , , and related papers. There is as well a certain connection with the Deligne work on the exceptional series of Lie groups, which is not understood yet.

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:

Theorem

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].

Show Proof

This is a heavy result, and we refer to and subsequent papers for the whole story, which involved the work of many people, all over the 2000s.

■

Next, in index [math]N=6[/math], 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 and the recent literature on the subject.

So long for small index. In higher index now, [math]N\in(4,\infty)[/math], where the Jones result in does apply, the precise correct “blowup” manipulation on the spectral measure is not known yet. Again, we refer here to , , and related papers.

Finally, one interesting question regards the case of large, uniform index, [math]N \gt \gt 0[/math]. Here the main examples are those coming from Theorem 16.35, with the underlying compact quantum group [math]G[/math] being assumed to be easy. But here, there is no need to do further probability, because we already did this, in chapters 13-14 above. \begin{exercises} Congratulations for having read this book, and no exercises for this final chapter. However, if looking for a good question, learn more, from Connes, Popa and others about the Murray-von Neumann hyperfinite factor [math]R[/math], and start doing some math, inside it. \begin{thebibliography}{99} \baselineskip=12.08pt \bibitem{afa}G.W. Anderson and B. Farrell, Asymptotically liberating sequences of random unitary matrices, Adv. Math. 255 (2014), 381--413. \bibitem{agz}G.W. Anderson, A. Guionnet and O. Zeitouni, An introduction to random matrices, Cambridge Univ. Press (2010). \bibitem{ans}M. 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General references

Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].