13a. Easy geometries

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

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We discuss here and in the next 3 chapters a number of more specialized questions, of algebraic, geometric, analytic and probabilistic nature. We will be interested in the main 9 examples of noncommutative geometries in our sense, which are as follows:

[[math]] \xymatrix@R=40pt@C=40pt{ \mathbb R^N_+\ar[r]&\mathbb T\mathbb R^N_+\ar[r]&\mathbb C^N_+\\ \mathbb R^N_*\ar[u]\ar[r]&\mathbb T\mathbb R^N_*\ar[u]\ar[r]&\mathbb C^N_*\ar[u]\\ \mathbb R^N\ar[u]\ar[r]&\mathbb T\mathbb R^N\ar[u]\ar[r]&\mathbb C^N\ar[u] } [[/math]]

Our purpose will be that of going beyond the basic level, where we are now, with a number of results regarding the coordinates [math]x_1,\ldots,x_N[/math] of such spaces:

• A first question, which is algebraic, is that of understanding the precise relations satisfied by these coordinates. We will see that this is related to the question of unifying the twisted and untwisted geometries, via intersection.
• A second question, which is analytic, is that of understanding the fixed [math]N[/math] behavior of these coordinates. This can be done via deformation methods. We will see as well that there is an unexpected link with quantum permutations.

Let us begin by discussing algebraic aspects. This is something quite fundamental. Indeed, in the classical case, the algebraic manifolds [math]X[/math] can be identified with the corresponding ideals of vanishing polynomials [math]J[/math], and the correspondence [math]X\leftrightarrow J[/math] is the foundation for all the known algebraic geometric theory, ancient or more modern.

In the free setting, things are in a quite primitive status, and a suitable theory of “noncommutative algebra”, useful in connection with our present considerations, is so far missing. Computing [math]J[/math] for the free spheres, and perhaps for some other spheres as well, is a problem which is difficult enough for us, and that we will investigate here.

As a starting point, we know that the above 9 geometries are easy, and looking in detail at this easiness property will be our first task. Let us first recall that we have:

Definition

A geometry [math](S,T,U,K)[/math] is called easy when [math]U,K[/math] are easy, and

[[math]] U=\{O_N,K\} [[/math]]

with the operation on the right being the easy generation operation.

To be more precise, in order for a geometry to be easy, the quantum groups [math]U,K[/math] must be of course easy, as stated above. Regarding now the generation condition, the point is that one of our general axioms for the nocommutative geometries, from chapter 4, states that we must have [math]U= \lt O_N,K \gt [/math], with the operation [math] \lt \,, \gt [/math] being a usual generation operation. And the above easy generation condition [math]U=\{O_N,K\}[/math] is something stronger, and so imposing this condition amounts in saying that we must have:

[[math]] \lt O_N,K \gt =\{O_N,K\} [[/math]]

The easy geometries in the above sense can be investigated by using:

Proposition

An easy geometry is uniquely determined by a pair [math](D,E)[/math] of categories of partitions, which must be as follows,

[[math]] \mathcal{NC}_2\subset D\subset P_2 [[/math]]

[[math]] \mathcal{NC}_{even}\subset E\subset P_{even} [[/math]]

and which are subject to the following intersection and generation conditions,

[[math]] D=E\cap P_2 [[/math]]

[[math]] E= \lt D,\mathcal{NC}_{even} \gt [[/math]]

and to the usual axioms for the associated quadruplet [math](S,T,U,K)[/math], where [math]U,K[/math] are respectively the easy quantum groups associated to the categories [math]D,E[/math].

Show Proof

This statement simply comes from the following conditions:

[[math]] U=\{O_N,K\} [[/math]]

[[math]] K=U\cap K_N^+ [[/math]]

To be more precise, let us look at Definition 13.1. The main condition there tells us that [math]U,K[/math] must be easy, coming from certain categories [math]D,E[/math]. It is clear that [math]D,E[/math] must appear as intermediate categories, as in the statement, and the fact that the intersection and generation conditions must be satisfied follows from:

[[math]] \begin{eqnarray*} U=\{O_N,K\}&\iff&D=E\cap P_2\\ K=U\cap K_N^+&\iff&E= \lt D,\mathcal{NC}_{even} \gt \end{eqnarray*} [[/math]]

Thus, we are led to the conclusion in the statement.

■

Generally speaking, the idea now is that, in the context of an easy geometry, everything can be reformulated in terms of the categories of partitions [math](D,E)[/math], which must satisfy the conditions in Proposition 13.2. Thus, we have in fact a diagram as follows:

[[math]] \xymatrix@R=30pt@C=30pt{ S\ar[rr]\ar[dr]\ar[dd]&&T\ar[ll]\ar[dd]\ar[dl]\\ &(D,E)\ar[ul]\ar[ur]\ar[dr]\ar[dl]\\ U\ar[uu]\ar[ur]\ar[rr]&&K\ar[ll]\ar[ul]\ar[uu] } [[/math]]

This is not suprising, because our main examples of geometries are the classical ones, governed by the commutation relations [math]ab=ba[/math], then the half-classical ones, coming from the half-commutation relations [math]abc=cba[/math], and then the free geometries, coming from no relations at all. Thus, modulo some technical conditions and axioms involving the quadruplets [math](S,T,U,K)[/math], which are there in order for our geometry to really “work”, everything comes down to the combinatorial structure which replaces the commutation relations [math]ab=ba[/math]. And the notion of category of partitions is precisely there for that.

This was for the idea. Now instead of discussing the full reformulation of our axions in terms of categories of partitions, which technically speaking will not bring many new things, let us work out at least the construction of the quadruplet [math](S,T,U,K)[/math]. In what regards the quantum groups, these come from via Tannakian duality, as follows:

Theorem

In the context of an easy geometry [math](S,T,U,K)[/math], we have:

[[math]] C(U)=C(U_N^+)\big/\left \lt T_\pi\in Hom(u^{\otimes k},u^{\otimes l})\Big|\forall k,l,\forall\pi\in D(k,l)\right \gt [[/math]]

Also, we have the following formula:

[[math]] C(K)=C(K_N^+)\big/\left \lt T_\pi\in Hom(u^{\otimes k},u^{\otimes l})\Big|\forall k,l,\forall\pi\in D(k,l)\right \gt [[/math]]

In fact, these formulae simply follow from the fact that [math]U[/math] is easy.

Show Proof

This follows from general easiness considerations. Indeed, the construction of the easy quantum groups in , based on the Tannakian duality of Woronowicz from , in its soft form from Malacarne , amounts in saying that the easy quantum group [math]G\subset U_N[/math] associated to a category of partitions [math]F=(F(k,l))[/math] is given by:

[[math]] C(G)=C(U_N^+)\big/\left \lt T_\pi\in Hom(u^{\otimes k},u^{\otimes l})\Big|\forall k,l,\forall\pi\in F(k,l)\right \gt [[/math]]

Thus, for the categories of partitions [math]D,E[/math] associated to an easy geometry, as in Proposition 13.2, the corresponding quantum groups are as follows:

[[math]] C(U)=C(U_N^+)\big/\left \lt T_\pi\in Hom(u^{\otimes k},u^{\otimes l})\Big|\forall k,l,\forall\pi\in D(k,l)\right \gt [[/math]]

[[math]] C(K)=C(U_N^+)\big/\left \lt T_\pi\in Hom(u^{\otimes k},u^{\otimes l})\Big|\forall k,l,\forall\pi\in E(k,l)\right \gt [[/math]]

But the first formula is the formula for [math]U[/math] in the statement. As for the second formula, this can be fine-tuned by using the following formula, again coming from easiness:

[[math]] C(K_N^+)=C(U_N^+)\big/\left \lt T_\pi\in Hom(u^{\otimes k},u^{\otimes l})\Big|\forall k,l,\forall\pi\in\mathcal{NC}_{even}(k,l)\right \gt [[/math]]

Indeed, by using the formula [math]E= \lt D,\mathcal{NC}_{even} \gt [/math] from Proposition 13.2, we have:

[[math]] C(K) =C(U_N^+)\big/\left \lt T_\pi\in Hom(u^{\otimes k},u^{\otimes l})\Big|\forall k,l,\forall\pi\in \lt D,\mathcal{NC}_{even} \gt (k,l)\right \gt [[/math]]

But constructing the algebra on the right amounts in dividing by the ideal coming from the partitions in [math]\mathcal{NC}_{even}[/math], which gives the algebra [math]C(K_N^+)[/math], and then further dividing by the ideal coming from the partitions in [math]D[/math], which gives the algebra in the statement.

■

Regarding now the associated torus [math]T[/math], which is not exactly covered by the easy quantum group formalism, the result here is a bit different, as follows:

Theorem

In the context of an easy geometry [math](S,T,U,K)[/math], we have:

[[math]] \Gamma=F_N\Big/\left \lt g_{i_1}\ldots g_{i_k}=g_{j_1}\ldots g_{j_l}\Big|\forall i,j,k,l,\exists\pi\in D(k,l),\delta_\pi\begin{pmatrix}i\\ j\end{pmatrix}\neq0\right \gt [[/math]]

In fact, this formula simply follows from the fact that [math]U[/math] is easy.

Show Proof

Let us denote by [math]g_i=u_{ii}[/math] the standard coordinates on the associated torus [math]T[/math], and consider the diagonal matrix formed by these coordinates:

[[math]] g=\begin{pmatrix}g_1\\&\ddots\\&&g_N\end{pmatrix} [[/math]]

We have the following computation:

[[math]] \begin{eqnarray*} C(T) &=&\left[C(U_N^+)\Big/\left \lt T_\pi\in Hom(u^{\otimes k},u^{\otimes l})\Big|\forall\pi\in D\right \gt \right]\Big/\left \lt u_{ij}=0\Big|\forall i\neq j\right \gt \\ &=&\left[C(U_N^+)\Big/\left \lt u_{ij}=0\Big|\forall i\neq j\right \gt \right]\Big/\left \lt T_\pi\in Hom(u^{\otimes k},u^{\otimes l})\Big|\forall\pi\in D\right \gt \\ &=&C^*(F_N)\Big/\left \lt T_\pi\in Hom(g^{\otimes k},g^{\otimes l})\Big|\forall\pi\in D\right \gt \end{eqnarray*} [[/math]]

Now observe that, with [math]g=diag(g_1,\ldots,g_N)[/math] as before, we have:

[[math]] T_\pi g^{\otimes k}(e_{i_1}\otimes\ldots\otimes e_{i_k})=\sum_{j_1\ldots j_l}\delta_\pi\begin{pmatrix}i_1&\ldots&i_k\\ j_1&\ldots&j_l\end{pmatrix}e_{j_1}\otimes\ldots\otimes e_{j_l}\cdot g_{i_1}\ldots g_{i_k} [[/math]]

On the other hand, we have as well:

[[math]] g^{\otimes l}T_\pi(e_{i_1}\otimes\ldots\otimes e_{i_k})=\sum_{j_1\ldots j_l}\delta_\pi\begin{pmatrix}i_1&\ldots&i_k\\ j_1&\ldots&j_l\end{pmatrix}e_{j_1}\otimes\ldots\otimes e_{j_l}\cdot g_{j_1}\ldots g_{j_l} [[/math]]

Thus, the commutation relation [math]T_\pi\in Hom(g^{\otimes k},g^{\otimes l})[/math] reads:

[[math]] \begin{eqnarray*} &&\sum_{j_1\ldots j_l}\delta_\pi\begin{pmatrix}i_1&\ldots&i_k\\ j_1&\ldots&j_l\end{pmatrix}e_{j_1}\otimes\ldots\otimes e_{j_l}\cdot g_{i_1}\ldots g_{i_k}\\ &=&\sum_{j_1\ldots j_l}\delta_\pi\begin{pmatrix}i_1&\ldots&i_k\\ j_1&\ldots&j_l\end{pmatrix}e_{j_1}\otimes\ldots\otimes e_{j_l}\cdot g_{j_1}\ldots g_{j_l} \end{eqnarray*} [[/math]]

Thus we obtain the formula in the statement, and the last assertion is clear.

■

Finally, regarding the sphere [math]S[/math], which is not a quantum group, but rather an homogeneous space, here the result is a bit more complicated, as follows:

Theorem

In the context of an easy geometry [math](S,T,U,K)[/math], we have

[[math]] C(S)=C(S^{N-1}_{\mathbb C,+})\Big/\left \lt x_{i_1}\ldots x_{i_k}=x_{j_1}\ldots x_{j_k}\Big|\forall i,j,k,l,\exists\pi\in D(k)\cap I_k,\delta_\pi\begin{pmatrix}i\\ j\end{pmatrix}\neq0\right \gt [[/math]]

where the set on the right, [math]I_k\subset P_2(k,k)[/math], is the set of colored permutations.

Show Proof

This follows indeed from Theorem 13.3, by applying the construction [math]U\to S[/math], which amounts in taking the first row space.

■

Summarizing, in the case of an easy geometry, we can reconstruct [math]S,T,U,K[/math] out of [math](D,E)[/math], or simply out of [math]D[/math], as done above. It is possible to reformulate everything in terms of [math](D,E)[/math], or just of [math]D[/math], by taking our axioms from chapter 4, and plugging in the formulae of [math]S,T,U,K[/math] in terms of [math](D,E)[/math], or in terms of [math]D[/math], coming from the above results. However, this remains something theoretical, and we will not get into details here.

General references

Banica, Teo (2024). "Affine noncommutative geometry". arXiv:2012.10973 [math.QA].