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Getting back to quantum groups, all this machinery is interesting for us. We will need the construction of the tensor and spin planar algebras [math]\mathcal T_N,\mathcal S_N[/math]. Let us start with:

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 [math]T_\pi:P_{k_1}\otimes\ldots\otimes P_{k_r}\to P_k[/math] being given by the formula

[[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 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 quantum group 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 16.22, is something elementary, in the same spirit as the verification of the functoriality properties of the correspondence [math]\pi\to T_\pi[/math], from easiness, 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 somehow as the “square root” of the tensor planar algebra [math]\mathcal T_N[/math]. Let us start with:

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 , 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 multiplinear maps [math]T_\pi:P_{k_1}\otimes\ldots\otimes P_{k_r}\to P_k[/math] being given by

[[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.

Here are some illustrating examples for the spin planar algebra calculus:

(1) 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]]

(2) 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]]

(3) 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]]

(4) 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]]

(5) 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:

[[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 then 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].

(6) 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_1i_2\ldots i_k})=e_{i_2\ldots i_ki_1} [[/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 fact:

Theorem

The multiplications, inclusions, expectations, Jones projections and rotations generate the set of all tangles, via the gluing operation.

Show Proof

This is something well-known and elementary, obtained by “chopping” the various planar tangles into small pieces, as in the above list. See .

■

Finally, in order for our discussion to be complete, we must talk as well about the [math]*[/math]-structure of the spin planar algebra. This is constructed 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]]

As before, we refer to Jones' paper for more on all this. Getting back now to quantum groups, following , we have the following result:

Theorem

Given [math]G\subset S_N^+[/math], consider the tensor powers of the associated coaction map on [math]C(X)[/math], where [math]X=\{1,\ldots,N\}[/math], which are the folowing linear maps:

[[math]] \Phi^k:C(X^k)\to C(X^k)\otimes C(G) [[/math]]

[[math]] e_{i_1\ldots i_k}\to\sum_{j_1\ldots j_k}e_{j_1\ldots j_k}\otimes u_{j_1i_1}\ldots u_{j_ki_k} [[/math]]

The fixed point spaces of these coactions, which are by definition the spaces

[[math]] P_k=\left\{ x\in C(X^k)\Big|\Phi^k(x)=1\otimes x\right\} [[/math]]

are given by [math]P_k=Fix(u^{\otimes k})[/math], and form a subalgebra of the spin planar algebra [math]\mathcal S_N[/math].

Show Proof

Since the map [math]\Phi[/math] is a coaction, its tensor powers [math]\Phi^k[/math] are coactions too, and at the level of fixed point algebras we have the following formula:

[[math]] P_k=Fix(u^{\otimes k}) [[/math]]

In order to prove now the planar algebra assertion, we will use Theorem 16.31. Consider the rotation [math]R_k[/math]. Rotating, then applying [math]\Phi^k[/math], and rotating backwards by [math]R_k^{-1}[/math] is the same as applying [math]\Phi^k[/math], then rotating each [math]k[/math]-fold product of coefficients of [math]\Phi[/math]. Thus the elements obtained by rotating, then applying [math]\Phi^k[/math], or by applying [math]\Phi^k[/math], then rotating, differ by a sum of Dirac masses tensored with commutators in [math]A=C(G)[/math]:

[[math]] \Phi^kR_k(x)-(R_k\otimes id)\Phi^k(x)\in C(X^k)\otimes [A,A] [[/math]]

Now let [math]\int_A[/math] be the Haar functional of [math]A[/math], and consider the conditional expectation onto the fixed point algebra [math]P_k[/math], which is given by the following formula:

[[math]] \phi_k=\left(id\otimes\int_A\right)\Phi^k [[/math]]

Since [math]\int_A[/math] is a trace, it vanishes on commutators. Thus [math]R_k[/math] commutes with [math]\phi_k[/math]:

[[math]] \phi_kR_k=R_k\phi_k [[/math]]

The commutation relation [math]\phi_kT=T\phi_l[/math] holds in fact for any [math](l,k)[/math]-tangle [math]T[/math]. These tangles are called annular, and the proof is by verification on generators of the annular category. In particular we obtain, for any annular tangle [math]T[/math]:

[[math]] \phi_kT\phi_l=T\phi_l [[/math]]

We conclude from this that the annular category is contained in the suboperad [math]\mathcal P'\subset\mathcal P[/math] of the planar operad consisting of tangles [math]T[/math] satisfying the following condition, where [math]\phi =(\phi_k)[/math], and where [math]i(.)[/math] is the number of input boxes:

[[math]] \phi T\phi^{\otimes i(T)}=T\phi^{\otimes i(T)} [[/math]]

On the other hand the multiplicativity of [math]\Phi^k[/math] gives [math]M_k\in\mathcal P'[/math]. Now since the planar operad [math]\mathcal P[/math] is generated by multiplications and annular tangles, it follows that we have [math]\mathcal P'=P[/math]. Thus for any tangle [math]T[/math] the corresponding multilinear map between spaces [math]P_k(X)[/math] restricts to a multilinear map between spaces [math]P_k[/math]. In other words, the action of the planar operad [math]\mathcal P[/math] restricts to [math]P[/math], and makes it a subalgebra of [math]\mathcal S_N[/math], as claimed.

■

As a second result now, also from , completing our study, we have:

Theorem

We have a bijection between quantum permutation groups and subalgebras of the spin planar algebra,

[[math]] (G\subset S_N^+)\quad\longleftrightarrow\quad (Q\subset\mathcal S_N) [[/math]]

given in one sense by the construction in Theorem 16.32, and in the other sense by a suitable modification of Tannakian duality.

Show Proof

The idea is that this will follow by applying Tannakian duality to the annular category over [math]Q[/math]. Let [math]n,m[/math] be positive integers. To any element [math]T_{n+m}\in Q_{n+m}[/math] we associate a linear map [math]L_{nm}(T_{n+m}):P_n(X)\to P_m(X)[/math] in the following way:

[[math]] L_{nm}\left(\begin{matrix}|\ |\ |\\ T_{n+m}\\ |\ |\ |\end{matrix}\right): \left(\begin{matrix}|\\ a_n\\ |\end{matrix}\right) \to \left(\begin{matrix} \hskip 1.5mm |\hskip 3.0mm |\hskip 3.0mm \cap\\ \ \ T_{n+m}\hskip 0.0mm |\\ \hskip 1.9mm |\hskip 1.2mm |\hskip 3.2mm |\hskip2.2mm |\\ a_n|\hskip 3.2mm |\hskip 2.2mm |\\ \hskip 2.1mm\cup \hskip3.5mm |\hskip 2.2mm | \end{matrix}\right) [[/math]]

That is, we consider the planar [math](n,n+m,m)[/math]-tangle having an small input [math]n[/math]-box, a big input [math]n+m[/math]-box and an output [math]m[/math]-box, with strings as on the picture of the right. This defines a certain multilinear map, as follows:

[[math]] P_n(X)\otimes P_{n+m}(X)\to P_m(X) [[/math]]

If we put the element [math]T_{n+m}[/math] in the big input box, we obtain in this way a certain linear map [math]P_n(X)\to P_m(X)[/math], that we call [math]L_{nm}[/math]. With this convention, let us set:

[[math]] Q_{nm}=\left\{ L_{nm}(T_{n+m}):P_n(X)\to P_m(X)\Big| T_{n+m}\in Q_{n+m}\right\} [[/math]]

These spaces form a Tannakian category, so by we obtain a Woronowicz algebra [math](A,u)[/math], such that the following equalities hold, for any [math]m,n[/math]:

[[math]] Hom(u^{\otimes m},u^{\otimes n})=Q_{mn} [[/math]]

We prove that [math]u[/math] is a magic unitary. We have [math]Hom(1,u^{\otimes 2})=Q_{02}=Q_2[/math], so the unit of [math]Q_2[/math] must be a fixed vector of [math]u^{\otimes 2}[/math]. But [math]u^{\otimes 2}[/math] acts on the unit of [math]Q_2[/math] as follows:

[[math]] \begin{eqnarray*} u^{\otimes 2}(1) &=&u^{\otimes 2}\left( \sum_i \begin{pmatrix}i&i\\ i&i\end{pmatrix}\right)\\ &=&\sum_{ikl}\begin{pmatrix}k&k\\ l&l\end{pmatrix}\otimes u_{ki}u_{li}\\ &=&\sum_{kl}\begin{pmatrix}k&k\\ l&l\end{pmatrix}\otimes (uu^t)_{kl} \end{eqnarray*} [[/math]]

From [math]u^{\otimes 2}(1)=1\otimes 1[/math] ve get that [math]uu^t[/math] is the identity matrix. Together with the unitarity of [math]u[/math], this gives the following formulae:

[[math]] u^t=u^*=u^{-1} [[/math]]

Consider the Jones projection [math]E_1\in Q_3[/math]. After isotoping, [math]L_{21}(E_1)[/math] looks as follows:

[[math]] L_{21}\left( \Bigl| \begin{matrix}\cup\\\cap\end{matrix}\right) : \begin{pmatrix} \,|\ |\\ {\ }^i_j{\ }^i_j\\ \,|\ |\end{pmatrix}\,\to\, \begin{pmatrix}\hskip -5.8mm |\\ {\ }^i_j{\ }^i_j\supset\\ \hskip -5.8mm |\end{pmatrix} =\,\delta_{ij}\begin{pmatrix}\,|\\ {\ }^i_i\\ \,|\end{pmatrix} [[/math]]

In other words, the linear map [math]M=L_{21}(E_1)[/math] is the multiplication [math]\delta_i\otimes\delta_j\to\delta_{ij}\delta_i[/math]:

[[math]] M\begin{pmatrix}i&i\\ j&j\end{pmatrix} =\delta_{ij}\begin{pmatrix}i\\ i\end{pmatrix} [[/math]]

In order to finish, consider the following element of [math]C(X)\otimes A[/math]:

[[math]] (M\otimes id)u^{\otimes 2}\left(\begin{pmatrix}i&i\\ j&j\end{pmatrix}\otimes 1\right) =\sum_k\begin{pmatrix}k\\ k\end{pmatrix}\delta_k\otimes u_{ki}u_{kj} [[/math]]

Since [math]M\in Q_{21}=Hom(u^{\otimes 2},u)[/math], this equals the following element of [math]C(X)\otimes A[/math]:

[[math]] u(M\otimes id)\left(\begin{pmatrix}i&i\\ j&j\end{pmatrix}\otimes 1\right) =\sum_k\begin{pmatrix}k\\ k\end{pmatrix}\delta_k\otimes\delta_{ij}u_{ki} [[/math]]

Thus we have [math]u_{ki}u_{kj}=\delta_{ij}u_{ki}[/math] for any [math]i,j,k[/math], which shows that [math]u[/math] is a magic unitary. Now if [math]P[/math] is the planar algebra associated to [math]u[/math], we have [math]Hom(1,v^{\otimes n})=P_n=Q_n[/math], as desired. As for the uniqueness, this is clear from the Peter-Weyl theory.

■

All the above might seem a bit technical, but is worth learning, and for good reason, because it is extremely powerful. As an example of application, if you agree with the bijection [math]G\leftrightarrow Q[/math] in Theorem 16.33, then [math]G=S_N^+[/math] itself, which is the biggest object on the left, must correspond to the smallest object on the right, namely [math]Q=TL_N[/math].

Back now to our usual business, graphs, we have the following result:

Theorem

The planar algebra associated to [math]G^+(X)[/math] is equal to the planar algebra generated by [math]d[/math], viewed as a [math]2[/math]-box in the spin planar algebra [math]\mathcal S_N[/math], with [math]N=|X|[/math].

Show Proof

We recall from the above that any quantum permutation group [math]G\subset S_N^+[/math] produces a subalgebra [math]P\subset\mathcal S_N[/math] of the spin planar algebra, given by:

[[math]] P_k=Fix(u^{\otimes k}) [[/math]]

In our case, the idea is that [math]G=G^+(X)[/math] comes via the relation [math]d\in End(u)[/math], but we can view this relation, via Frobenius duality, as a relation of the following type:

[[math]] \xi_d\in Fix(u^{\otimes 2}) [[/math]]

Indeed, let us view the adjacency matrix [math]d\in M_N(0,1)[/math] as a 2-box in [math]\mathcal S_N[/math], by using the canonical identification between [math]M_N(\mathbb C)[/math] and the algebra of 2-boxes [math]\mathcal S_N(2)[/math]:

[[math]] (d_{ij})\leftrightarrow \sum_{ij} d_{ij}\begin{pmatrix}i&i\\ j&j\end{pmatrix} [[/math]]

Let [math]P[/math] be the planar algebra associated to [math]G^+(X)[/math] and let [math]Q[/math] be the planar algebra generated by [math]d[/math]. The action of [math]u^{\otimes 2}[/math] on [math]d[/math] viewed as a 2-box is given by:

[[math]] u^{\otimes 2}\left(\sum_{ij} d_{ij}\begin{pmatrix}i&i\\ j&j\end{pmatrix}\right) =\sum_{ijkl} d_{ij}\begin{pmatrix}k&k\\ l&l\end{pmatrix}\otimes u_{ki}u_{lj} =\sum_{kl}\begin{pmatrix}k&k\\ l&l\end{pmatrix}\otimes (udu^t)_{kl} [[/math]]

Since [math]v[/math] is a magic unitary commuting with [math]d[/math] we have:

[[math]] udu^t=duu^t=d [[/math]]

But this means that [math]d[/math], viewed as a 2-box, is in the algebra [math]P_2[/math] of fixed points of [math]u^{\otimes 2}[/math]. Thus [math]Q\subset P[/math]. As for [math]P\subset Q[/math], this follows from the duality found above.

■

Generally speaking, the above material, when coupled with what we did in this book about graphs, leads us into the classification of the subalgebras of the spin planar algebra generated by a 2-box. But this can be regarded as a particular case of the Bisch-Jones question of classifying, in general, the planar algebras generated by a 2-box .

There are many more things that can be said here, notably with the introduction of a general notion of “quantum graph”, which is something quite interesting on its own, and which brings the whole graph problematics closer to the level of generality of . For more on all this, we refer to the book , which is more specialized.

Finally, let us mention that, as already explained since the beginning of this book, graphs are not everything, in relation with discrete mathematics. Discrete mathematics is something far wider than graph theory, with all sorts of interesting objects involved, which are not necessarily graphs. But the main principles that we learned here, namely that things are usually encoded by a matrix, and that symmetries and quantum symmetries play a key role, generally apply. For more on all this, other aspects of discrete mathematics, and related algebraic techniques, you can have a look at , , . \begin{exercises} Congratulations for having read this book, and no exercises for this final chapter. However, for further reading, you have many possible books, on graphs and related topics. We have referenced some below, and in the hope that you will like some of them. \begin{thebibliography}{99} \baselineskip=14pt \bibitem{ar1}V.I. 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General references

Banica, Teo (2024). "Graphs and their symmetries". arXiv:2406.03664 [math.CO].