14a. Planar algebras

[math] \newcommand{\mathds}{\mathbb}[/math]

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We have seen the foundations of subfactor theory, and the main examples of subfactors, all having integer index. Following Jones' paper , in this chapter we go into the core of the theory, with the idea in mind of axiomatizing the combinatorics of a given subfactor [math]A_0\subset A_1[/math], via an object similar to the tensor categories for the quantum groups.

So, our starting point will be an arbitrary subfactor [math]A_0\subset A_1[/math], assumed to have finite index, [math][A_1:A_0] \lt \infty[/math]. Let us first review first what can be said about it, by using the Jones basic construction. We recall from chapter 13 that we have the following result:

Theorem

Given an inclusion of [math]{\rm II}_1[/math] factors [math]A_0\subset A_1[/math], with Jones tower

[[math]] A_0\subset_{e_1}A_1\subset_{e_2}A_2\subset_{e_3}A_3\subset\ldots\ldots [[/math]]

the sequence of projections [math]e_1,e_2,e_3,\ldots\in B(H)[/math] produces a representation

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

of the Temperley-Lieb algebra of index [math]N=[A_1:A_0][/math]. Moreover, we have

[[math]] TL_N\subset P [[/math]]

where [math]P=(P_k)[/math] is the graded algebra formed by the commutants [math]P_k=A_0'\cap A_k[/math].

Show Proof

There are two statements here, both due to Jones , that we know from chapter 13 above, the idea for this, in short, being as follows:

(1) A detailed study of the basic construction, performed in chapter 13, shows that the rescaled sequence of Jones projections [math]e_1,e_2,e_3,\ldots\in B(H)[/math] behaves algebrically exactly as the sequence of standard generators [math]\varepsilon_1,\varepsilon_2,\varepsilon_3,\ldots\in TL_N[/math]. Thus we have an embedding [math]TL_N\subset B(H)[/math], where [math]H[/math] is the Hilbert space where our subfactor [math]A_0\subset A_1[/math] lives.

(2) Once again by carefully looking at the Jones basic construction, the more precise conclusion is that the image of the representation [math]TL_N\subset B(H)[/math] constructed above lives indeed in the graded algebra [math]P=(P_k)[/math] formed by the commutants [math]P_k=A_0'\cap A_k[/math].

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Quite remarkably, the planar algebra structure of [math]TL_N[/math], taken in an intuitive sense, that of composing planar diagrams, extends to a planar algebra structure of [math]P[/math]. In order to discuss this key result, let us start with the axioms for planar algebras. Following Jones' paper , we have the following definition:

Definition

The planar algebras are defined as follows:

• We consider rectangles in the plane, with the sides parallel to the coordinate axes, and taken up to planar isotopy, and we call such rectangles boxes.
• A labeled box is a box with [math]2k[/math] marked points on its boundary, [math]k[/math] on its upper side, and [math]k[/math] on its lower side, for some integer [math]k\in\mathbb N[/math].
• A tangle is labeled box, containing a number of labelled boxes, with all marked points, on the big and small boxes, being connected by noncrossing strings.
• A planar algebra is a sequence of finite dimensional vector spaces [math]P=(P_k)[/math], together with linear maps [math]P_{k_1}\otimes\ldots\otimes P_{k_r}\to P_k[/math], one for each tangle, such that the gluing of tangles corresponds to the composition of linear maps.

In this definition we are using rectangles, but everything being up to isotopy, we could have used instead circles with marked points, as in , which is the same thing. Our choice for using rectangles comes from the main examples that we have in mind, to be discussed below, where the planar algebra structure is best viewed by using rectangles.

Let us also mention that Definition 14.2 is something quite simplified, based on . As explained in , in order for subfactors to produce planar algebras and vice versa, there are quite a number of supplementary axioms that must be added, and in view of this, it is perhaps better to start with something stronger than Definition 14.2, as basic axioms. However, as before with rectangles vs circles, our axiomatic choices here are mainly motivated by the concrete examples that we have in mind.

As a basic example of a planar algebra, we have the Temperley-Lieb algebra:

Theorem

The Temperley-Lieb algebra [math]TL_N[/math], viewed as sequence of finite dimensional vector spaces

[[math]] TL_N=(TL_N(k))_{k\in\mathbb N} [[/math]]

is a planar algebra in the above sense, with the corresponding linear maps associated to the planar tangles

[[math]] TL_N(k_1)\otimes\ldots\otimes TL_N(k_r)\to TL_N(k) [[/math]]

appearing by putting the various [math]TL_N(k_i)[/math] diagrams into the small boxes of the given tangle, which produces a [math]TL_N(k)[/math] diagram.

Show Proof

This is something trivial, which follows from definitions:

(1) Assume indeed that we are given a planar tangle [math]\pi[/math] in the sense of Definition 14.2, consisting of a box having [math]2k[/math] marked points on its boundary, and containing [math]r[/math] small boxes, having respectively [math]2k_1,\ldots,2k_r[/math] marked points on their boundaries, and then a total of [math]k+\Sigma k_i[/math] noncrossing strings, connecting the various [math]2k+\Sigma 2k_i[/math] marked points.

(2) We want to associate to this planar tangle [math]\pi[/math] a linear map as follows:

[[math]] T_\pi:TL_N(k_1)\otimes\ldots\otimes TL_N(k_r)\to TL_N(k) [[/math]]

For this purpose, by linearity, it is enough to construct elements as follows, for any choice of Temperley-Lieb diagrams [math]\sigma_i\in TL_N(k_i)[/math], with [math]i=1,\ldots,r[/math]:

[[math]] T_\pi(\sigma_1\otimes\ldots\otimes\sigma_r)\in TL_N(k) [[/math]]

(3) But constructing such an element is obvious, simply by putting the various diagrams [math]\sigma_i\in TL_N(k_i)[/math] into the small boxes the given tangle [math]\pi[/math]. Indeed, this procedure produces a certain diagram in [math]TL_N(k)[/math], that we can call [math]T_\pi(\sigma_1\otimes\ldots\otimes\sigma_r)[/math], as above.

(4) Finally, we still have to check that everything is well-defined up to planar isotopy, and that the gluing of tangles corresponds to the composition of linear maps. But both these checks are trivial, coming from the definition of [math]TL_N[/math], and we are done.

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As a conclusion, [math]P=TL_N[/math] is indeed a planar algebra, and of somewhat “trivial” type, with the triviality coming from the fact that, in this case, the elements of [math]P[/math] are planar diagrams themselves, and so the planar structure appears trivially. The Temperley-Lieb planar algebra [math]TL_N[/math] is however an important planar algebra, because it is the “smallest” one, appearing inside the planar algebra of any subfactor. But more on this later, when talking about the relation between planar algebras and subfactors.

Moving ahead, here is our second basic example of a planar algebra, due to Bisch-Jones , which is still “trivial” in the above sense, with the elements of the planar algebra being planar diagrams themselves, but which appears in a more complicated way:

Theorem

The Fuss-Catalan algebra [math]FC_{N,M}[/math], which appears by coloring the Temperley-Lieb diagrams with black/white colors, clockwise, as follows

[[math]] \circ\bullet\bullet\circ\circ\bullet\bullet\circ\ldots\ldots\ldots\circ\bullet\bullet\circ [[/math]]

and keeping those diagrams whose strings connect either [math]\circ-\circ[/math] or [math]\bullet-\bullet[/math], is a planar algebra, with again the corresponding linear maps associated to the planar tangles

[[math]] FC_{N,M}(k_1)\otimes\ldots\otimes FC_{N,M}(k_r)\to FC_{N,M}(k) [[/math]]

appearing by putting the various [math]FC_{N,M}(k_i)[/math] diagrams into the small boxes of the given tangle, which produces a [math]FC_{N,M}(k)[/math] diagram.

Show Proof

The proof here is nearly identical to the proof of Theorem 14.3, with the only change appearing at the level of the colors. To be more precise:

(1) Forgetting about upper and lower sequences of points, which must be joined by strings, a Temperley-Lieb diagram can be thought of as being just a collection of strings, say black strings, which compose in the obvious way, with the rule that the value of the circle, which is now a black circle, is [math]N[/math]. And it is this obvious composition rule that gives the planar algebra structure, as explained in the proof of Theorem 14.3.

(2) Similarly, forgetting about sequences of points, a Fuss-Catalan diagram can be thought of as being a collection of strings, which come now in two colors, black and white. These Fuss-Catalan diagrams compose in the obvious way, with the rule that the value of the black circle is [math]N[/math], and the value of the white circle is [math]M[/math]. And it is this obvious composition rule that gives the planar algebra structure, as before for [math]TL_N[/math].

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The same comments as those for [math]TL_N[/math] apply. On one hand, [math]FC_{N,M}[/math] is by definition a “trivial” planar algebra, with the triviality coming from the fact that its elements are planar diagrams themselves. On the other hand, [math]FC_{N,M}[/math] is an important planar algebra, because it can be shown to appear inside the planar algebra of any subfactor [math]A\subset B[/math], assuming that an intermediate subfactor [math]A\subset C\subset B[/math] is present. But more on this later, when talking about the relation between planar algebras and subfactors.

Getting back now to generalities, and to Definition 14.2, that of a general planar algebra, we have so far two illustrations for it, which, while both important, are both “trivial”, with the planar structure simply coming from the fact that, in both these illustrations, the elements of the planar algebra are planar diagrams themselves.

In general, the planar algebras are more complicated than this, and we will see some further examples in a moment. However, the idea is very simple, namely “the elements of a planar algebra are not necessarily diagrams, but they behave like diagrams”.

Let us begin with the construction of the tensor planar algebra [math]\mathcal T_N[/math], which is the third most important planar algebra, in our series of examples. This algebra is as follows:

Definition

The tensor planar algebra [math]\mathcal T_N[/math] is the sequence of vector spaces

[[math]] P_k=M_N(\mathbb C)^{\otimes k} [[/math]]

with the multilinear maps associated to the various [math]k[/math]-tangles

[[math]] T_\pi:P_{k_1}\otimes\ldots\otimes P_{k_r}\to P_k [[/math]]

being given by the following formula, in multi-index notation,

[[math]] T_\pi(e_{i_1}\otimes\ldots\otimes e_{i_r})=\sum_j\delta_\pi(i_1,\ldots,i_r:j)e_j [[/math]]

with the Kronecker symbols [math]\delta_\pi[/math] being [math]1[/math] if the indices fit, and being [math]0[/math] otherwise.

In other words, we are using here a construction which is very similar to the construction [math]\pi\to T_\pi[/math] from easy quantum groups. We put the indices of the basic tensors on the marked points of the small boxes, in the obvious way, and the coefficients of the output tensor are then given by Kronecker symbols, exactly as in the easy case.

The fact that we have indeed a planar algebra, in the sense that the gluing of tangles corresponds to the composition of linear maps, as required by Definition 14.2, is something elementary, in the same spirit as the verification of the functoriality properties of the correspondence [math]\pi\to T_\pi[/math], discussed in chapter 8, and we refer here to Jones .

Let us discuss now a second planar algebra of the same type, which is important as well for various reasons, namely the spin planar algebra [math]\mathcal S_N[/math]. This planar algebra appears somewhat as a “square root” of the tensor planar algebra [math]\mathcal T_N[/math], and its construction is quite similar, but by using this time the algebra [math]\mathbb C^N[/math] instead of the algebra [math]M_N(\mathbb C)[/math].

There is one subtlety, however, coming from the fact that the general planar algebra formalism, from Definition 14.2 above, requires the tensors to have even length. Note that this was automatic for [math]\mathcal T_N[/math], where the tensors of [math]M_N(\mathbb C)[/math] have length 2. In the case of the spin planar algebra [math]\mathcal S_N[/math], we want the vector spaces to be:

[[math]] P_k=(\mathbb C^N)^{\otimes k} [[/math]]

Thus, we must double the indices of the tensors, in the following way:

Definition

We write the standard basis of [math](\mathbb C^N)^{\otimes k}[/math] in [math]2\times k[/math] matrix form,

[[math]] e_{i_1\ldots i_k}= \begin{pmatrix}i_1 & i_1 &i_2&i_2&i_3&\ldots&\ldots\\ i_k&i_k&i_{k-1}&\ldots&\ldots&\ldots&\ldots \end{pmatrix} [[/math]]

by duplicating the indices, and then writing them clockwise, starting from top left.

Now with this convention in hand for the tensors, we can formulate the construction of the spin planar algebra [math]\mathcal S_N[/math], also from Jones , as follows:

Definition

The spin planar algebra [math]\mathcal S_N[/math] is the sequence of vector spaces

[[math]] P_k=(\mathbb C^N)^{\otimes k} [[/math]]

written as above, with the multilinear maps associated to the various [math]k[/math]-tangles

[[math]] T_\pi:P_{k_1}\otimes\ldots\otimes P_{k_r}\to P_k [[/math]]

being given by the following formula, in multi-index notation,

[[math]] T_\pi(e_{i_1}\otimes\ldots\otimes e_{i_r})=\sum_j\delta_\pi(i_1,\ldots,i_r:j)e_j [[/math]]

with the Kronecker symbols [math]\delta_\pi[/math] being [math]1[/math] if the indices fit, and being [math]0[/math] otherwise.

In other words, we are using exactly the same construction as for the tensor planar algebra [math]\mathcal T_N[/math], which was itself very related to the easy quantum group formalism, but with [math]M_N(\mathbb C)[/math] replaced by [math]\mathbb C^N[/math], with the indices doubled, as in Definition 14.6. As before with the tensor planar algebra [math]\mathcal T_N[/math], the fact that the spin planar algebra [math]\mathcal S_N[/math] is indeed a planar algebra is something rather trivial, coming from definitions.

Observe however that, unlike our previous planar algebras [math]TL_N[/math] and [math]FC_{N,M}[/math], which were “trivial” planar algebras, their elements being planar diagrams themselves, the planar algebras [math]\mathcal T_N[/math] and [math]\mathcal S_N[/math] are not trivial, their elements being not exactly planar diagrams. Let us also mention that the planar algebras [math]\mathcal T_N[/math] and [math]\mathcal S_N[/math] are important for a number of reasons, in the context of the fixed point subfactors, to be discussed later on.

Getting back now to the planar algebra structure of [math]\mathcal T_N[/math] and [math]\mathcal S_N[/math], which is something quite fundamental, worth being well understood, let us have here some more discussion. Generally speaking, the planar calculus for tensors is quite simple, and does not really require diagrams. Indeed, it suffices to imagine that the way various indices appear, travel around and dissapear is by following some obvious strings connecting them. Here are some illustrations for this principle, for the spin planar algebra [math]\mathcal S_N[/math]: Example

Identity, multiplication, inclusion.

The identity [math]1_k[/math] is the [math](k,k)[/math]-tangle having vertical strings only. The solutions of [math]\delta_{1_k}(x,y)=1[/math] being the pairs of the form [math](x,x)[/math], this tangle [math]1_k[/math] acts by the identity:

[[math]] 1_k\begin{pmatrix}j_1 & \ldots & j_k\\ i_1 & \ldots & i_k\end{pmatrix}=\begin{pmatrix}j_1 & \ldots & j_k\\ i_1 & \ldots & i_k\end{pmatrix} [[/math]]

The multiplication [math]M_k[/math] is the [math](k,k,k)[/math]-tangle having 2 input boxes, one on top of the other, and vertical strings only. It acts in the following way:

[[math]] M_k\left( \begin{pmatrix}j_1 & \ldots & j_k\\ i_1 & \ldots & i_k\end{pmatrix} \otimes\begin{pmatrix}l_1 & \ldots & l_k\\ m_1 & \ldots & m_k\end{pmatrix} \right)= \delta_{j_1m_1}\ldots \delta_{j_km_k} \begin{pmatrix}l_1 & \ldots & l_k\\ i_1 & \ldots & i_k\end{pmatrix} [[/math]]

The inclusion [math]I_k[/math] is the [math](k,k+1)[/math]-tangle which looks like [math]1_k[/math], but has one more vertical string, at right of the input box. Given [math]x[/math], the solutions of [math]\delta_{I_k}(x,y)=1[/math] are the elements [math]y[/math] obtained from [math]x[/math] by adding to the right a vector of the form [math](^l_l)[/math], and so:

[[math]] {I_k}\begin{pmatrix}j_1 & \ldots & j_k\\ i_1 & \ldots & i_k\end{pmatrix}= \sum_l\begin{pmatrix}j_1 & \ldots & j_k& l\\ i_1 & \ldots & i_k& l\end{pmatrix} [[/math]]

Observe that [math]I_k[/math] is an inclusion of algebras, and that the various [math]I_k[/math] are compatible with each other. The union of the algebras [math]\mathcal S_N(k)[/math] is a graded algebra, denoted [math]\mathcal S_N[/math].

Along the same lines, some other important tangles are as follows: Example

Expectation, Jones projection, trace.

The expectation [math]U_k[/math] is the [math](k+1,k)[/math]-tangle which looks like [math]1_k[/math], but has one more string, connecting the extra 2 input points, both at right of the input box:

[[math]] U_k \begin{pmatrix}j_1 & \ldots &j_k& j_{k+1}\\ i_1 & \ldots &i_k& i_{k+1}\end{pmatrix}= \delta_{i_{k+1}j_{k+1}} \begin{pmatrix}j_1 & \ldots & j_k\\ i_1 & \ldots & i_k\end{pmatrix} [[/math]]

Observe that [math]U_k[/math] is a bimodule morphism with respect to [math]I_k[/math].

The Jones projection [math]E_k[/math] is a [math](0,k+2)[/math]-tangle, having no input box. There are [math]k[/math] vertical strings joining the first [math]k[/math] upper points to the first [math]k[/math] lower points, counting from left to right. The remaining upper 2 points are connected by a semicircle, and the remaining lower 2 points are also connected by a semicircle. We have the following formula:

[[math]] E_k(1)=\sum_{ijl}\begin{pmatrix}i_1 & \ldots &i_k&j&j\\ i_1 & \ldots &i_k&l&l\end{pmatrix} [[/math]]

The elements [math]e_k=N^{-1}E_k(1)[/math] are projections, and define a representation of the infinite Temperley-Lieb algebra of index [math]N[/math] inside the inductive limit algebra [math]\mathcal S_N[/math].

The trace [math]T_k[/math] is the [math](k,0)[/math] tangle which “closes the diagram”, by connecting upper points with lower points with noncrossing strings at right, in the obvious way:

[[math]] T_k \begin{pmatrix}j_1 & \ldots &j_k\\ i_1 & \ldots &i_k\end{pmatrix}= \delta_{i_1j_1}\ldots\delta_{i_kj_k} [[/math]]

This tangle implements a trace on the planar algebra, and the expectations [math]U_k[/math] constructed above are then the conditional expectations with respect to this trace.

Finally, again along the same lines, we have the following two key tangles:

Example

Rotation, shift.

The rotation [math]R_k[/math] is the [math](k,k)[/math]-tangle which looks like [math]1_k[/math], but the first 2 input points are connected to the last 2 output points, and the same happens at right:

[[math]] R_k=\begin{matrix} \hskip 0.3mm\Cap \ |\ |\ |\ |\hskip -0.5mm |\cr |\hskip -0.5mm |\hskip 10.3mm |\hskip -0.5mm |\cr \hskip -0.3mm|\hskip -0.5mm |\ |\ |\ |\ \hskip -0.1mm\Cup \end{matrix} [[/math]]

The action of [math]R_k[/math] on the standard basis is by rotation of the indices, as follows:

[[math]] R_k(e_{i_1\ldots i_k})=e_{i_2i_3\ldots i_ki_1} [[/math]]

Thus [math]R_k[/math] acts by an order [math]k[/math] linear automorphism of [math]\mathcal S_N(k)[/math], also called rotation.

As for the shift [math]S_k[/math], this is the [math](k,k+2)[/math]-tangle which looks like [math]1_k[/math], but has two more vertical strings, at left of the input box. This tangle acts as follows:

[[math]] S_k\begin{pmatrix}j_1 & \ldots & j_k\\ i_1 & \ldots & i_k\end{pmatrix}= \sum_{lm}\begin{pmatrix}l&m&j_1&\ldots&j_k\\l&m&i_1&\ldots&i_k\end{pmatrix} [[/math]]

Observe that [math]S_k[/math] is an inclusion of algebras, which is different from [math]I_{k+1}I_k[/math].

There are many other interesting examples of [math]k[/math]-tangles, but in view of our present purposes, we can actually stop here, due to the following key fact, which basically reduces everything to the study of the above particular tangles, and that we will use many times in what follows, for the various planar algebra results that we will prove:

Theorem

The following tangles generate the set of all tangles, via gluing:

• Multiplications, inclusions.
• Expectations, Jones projections.
• Rotations or shifts.

Show Proof

As a first observation, the tangles in the statement are exactly those in the above examples, with the identity and trace tangles removed, due to the fact that these tangles won't bring anything new. Also, the statement itself consists in fact of 2 statements, depending on whether rotations and shifts are chosen in (3), with this being something technical, coming from the fact that we will need in what follows both these 2 statements. As for the proof, this is something elementary, obtained by “chopping” the various planar tangles into small pieces, belonging to the above list. See Jones .

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Finally, in order for things to be complete, we must talk as well about the [math]*[/math]-structure. Once again this is constructed as in the easy quantum group calculus, as follows:

[[math]] \begin{pmatrix}j_1 & \ldots & j_k\\ i_1 & \ldots & i_k\end{pmatrix}^* =\begin{pmatrix}i_1 & \ldots & i_k\\ j_1 & \ldots & j_k\end{pmatrix} [[/math]]

Summarizing, the sequence of vector spaces [math]\mathcal S_N(k)=(\mathbb C^N)^{\otimes k}[/math] has a planar [math]*[/math]-algebra structure, called spin planar algebra of index [math]N=|X|[/math]. See Jones .

As a conclusion to all this, we have so far an abstract definition for the planar algebras, then two very basic examples, namely [math]TL_N[/math] and [math]FC_{N,M}[/math], where the elements of the planar algebra are actual diagrams, composing as the diagrams do, by gluing, and then two examples which are slightly more complicated, namely [math]\mathcal T_N[/math] and [math]\mathcal S_N[/math], where the planar algebra elements are tensors, composing according to the usual rules for the tensors.

General references

Banica, Teo (2024). "Principles of operator algebras". arXiv:2208.03600 [math.OA].