13c. Twists, intersections

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Our purpose now will be that of going beyond the above results, with a number of more specialized results regarding the coordinates [math]x_1,\ldots,x_N[/math] of our real spheres. To be more precise, a first question that we would like to solve, which is of purely algebraic nature, is that of understanding the precise relations satisfied by these coordinates [math]x_1,\ldots,x_N[/math] over our real spheres. We will see, in a somewhat unexpected way, that this is related to the question of unifying the twisted and untwisted geometries, via intersection.

Let us begin by recalling the construction of the twisted real spheres, which was discussed in chapter 11. This is something very simple, as follows:

Definition

The subspheres [math]\bar{S}^{N-1}_\mathbb R,\bar{S}^{N-1}_{\mathbb R,*}\subset S^{N-1}_{\mathbb R,+}[/math] are constructed by imposing the following conditions on the standard coordinates [math]x_1,\ldots,x_N[/math]:

• [math]\bar{S}^{N-1}_\mathbb R[/math]: [math]x_ix_j=-x_jx_i[/math], for any [math]i\neq j[/math].
• [math]\bar{S}^{N-1}_{\mathbb R,*}[/math]: [math]x_ix_jx_k=-x_kx_jx_i[/math] for any [math]i,j,k[/math] distinct, [math]x_ix_jx_k=x_kx_jx_i[/math] otherwise.

Here the fact that we have indeed [math]\bar{S}^{N-1}_\mathbb R\subset\bar{S}^{N-1}_{\mathbb R,*}[/math] comes from the following computations, for [math]a,b,c\in\{x_i\}[/math] distinct, where [math]x_1,\ldots,x_N[/math] are the standard coordinates on [math]\bar{S}^{N-1}_\mathbb R[/math]:

[[math]] abc=-bac=bca=-cba [[/math]]

[[math]] aab=-aba=baa [[/math]]

Summarizing, we have a total of 5 real spheres, or rather a total of [math]3+3=6[/math] real spheres, with the convention that the free real sphere equals its twist:

[[math]] S^{N-1}_{\mathbb R,+}=\bar{S}^{N-1}_{\mathbb R,+} [[/math]]

The point now is that we can intersect these [math]3+3=6[/math] spheres, and we end up with a total of [math]3\times3=9[/math] real spheres, in a generalized sense, as follows:

Definition

Associated to any integer [math]N\in\mathbb N[/math] are the generalized spheres

[[math]] \xymatrix@R=13mm@C=3mm{ S^{N-1}_\mathbb R\ar[rr]&&S^{N-1}_{\mathbb R,*}\ar[rrr]&&&S^{N-1}_{\mathbb R,+}\\ S^{N-1}_\mathbb R\cap\bar{S}^{N-1}_{\mathbb R,*}\ar[rr]\ar[u]&&S^{N-1}_{\mathbb R,*}\cap\bar{S}^{N-1}_{\mathbb R,*}\ar[rrr]\ar[u]&&&\bar{S}^{N-1}_{\mathbb R,*}\ar[u]\\ S^{N-1}_\mathbb R\cap\bar{S}^{N-1}_\mathbb R\ar[rr]\ar[u]&&S^{N-1}_{\mathbb R,*}\cap\bar{S}^{N-1}_\mathbb R\ar[rrr]\ar[u]&&&\bar{S}^{N-1}_\mathbb R\ar[u]} [[/math]]

obtained by intersecting the [math]3[/math] twisted real spheres and the [math]3[/math] untwisted real spheres.

In order to compute the various intersections appearing above, which in general cannot be thought of as being smooth, let us introduce the following objects:

Definition

The polygonal spheres are real algebraic manifolds, defined as

[[math]] S^{N-1,d-1}_\mathbb R=\left\{x\in S^{N-1}_\mathbb R\Big|x_{i_0}\ldots x_{i_d}=0,\forall i_0,\ldots,i_d\ {\rm distinct}\right\} [[/math]]

depending on integers [math]1\leq d\leq N[/math].

These spheres, introduced and studied in , are not smooth in general, but recall that we are currently doing algebraic geometry, rather than differential geometry, and with actually the colorful name “polygonal spheres”, used in and that we will use here too, being there for reminding us that. To be more precise, the point is that the problem that we want to solve, namely understanding the precise relations satisfied by the coordinates [math]x_1,\ldots,x_N[/math] for the real spheres, naturally leads into polygonal spheres.

More generally now, we have the following construction of “generalized polygonal spheres”, which applies to the half-classical and twisted cases too:

[[math]] C(\dot{S}^{N-1,d-1}_{\mathbb R,\times})=C\big(\dot{S}^{N-1}_{\mathbb R,\times}\big)\Big/\Big \lt x_{i_0}\ldots x_{i_d}=0,\forall i_0,\ldots,i_d\ {\rm distinct}\Big \gt [[/math]]

Here the fact that in the classical case we obtain the polygonal spheres from Definition 13.16 comes from a straightforward application of the Gelfand theorem.

With these conventions, we have the following result, dealing with all the spheres that we have so far in real case, namely twisted, untwisted and intersections:

Theorem

The diagram obtained by intersecting the twisted and untwisted real spheres, from Definition 13.15, is given by

[[math]] \xymatrix@R=13mm@C=16mm{ S^{N-1}_\mathbb R\ar[r]&S^{N-1}_{\mathbb R,*}\ar[r]&S^{N-1}_{\mathbb R,+}\\ S^{N-1,1}_\mathbb R\ar[r]\ar[u]&S^{N-1,1}_{\mathbb R,*}\ar[r]\ar[u]&\bar{S}^{N-1}_{\mathbb R,*}\ar[u]\\ S^{N-1,0}_\mathbb R\ar[r]\ar[u]&\bar{S}^{N-1,1}_\mathbb R\ar[r]\ar[u]&\bar{S}^{N-1}_\mathbb R\ar[u]} [[/math]]

and so all these spheres are generalized polygonal spheres.

Show Proof

Consider the 4-diagram obtained by intersecting the 5 main spheres:

[[math]] \xymatrix@R=13mm@C=13mm{ S^{N-1}_\mathbb R\cap\bar{S}^{N-1}_{\mathbb R,*}\ar[r]&S^{N-1}_{\mathbb R,*}\cap\bar{S}^{N-1}_{\mathbb R,*}\\ S^{N-1}_\mathbb R\cap\bar{S}^{N-1}_\mathbb R\ar[r]\ar[u]&S^{N-1}_{\mathbb R,*}\cap\bar{S}^{N-1}_\mathbb R\ar[u]} [[/math]]

We must prove that this diagram coincides with the 4-diagram appearing at bottom left in the statement, which is as follows:

[[math]] \xymatrix@R=13mm@C=13mm{ S^{N-1,1}_\mathbb R\ar[r]&S^{N-1,1}_{\mathbb R,*}\\ S^{N-1,0}_\mathbb R\ar[r]\ar[u]&\bar{S}^{N-1,1}_\mathbb R\ar[u]} [[/math]]

But this is clear, because combining the commutation and anticommutation relations leads to the vanishing relations defining the spheres of type [math]\dot{S}^{N-1,d-1}_{\mathbb R,\times}[/math]. More precisely:

(1) [math]S^{N-1}_\mathbb R\cap\bar{S}^{N-1}_\mathbb R[/math] consists of the points [math]x\in S^{N-1}_\mathbb R[/math] such that, for any [math]i\neq j[/math]:

[[math]] x_ix_j=-x_jx_i [[/math]]

Now since we have as well [math]x_ix_j=x_jx_i[/math], for any [math]i,j[/math], this relation reads [math]x_ix_j=0[/math] for [math]i\neq j[/math], which means that we have [math]x\in S^{N-1,0}_\mathbb R[/math], as desired.

(2) [math]S^{N-1}_\mathbb R\cap\bar{S}^{N-1}_{\mathbb R,*}[/math] consists of the points [math]x\in S^{N-1}_\mathbb R[/math] such that, for [math]i,j,k[/math] distinct:

[[math]] x_ix_jx_k=-x_kx_jx_i [[/math]]

Once again by commutativity, this relation is equivalent to [math]x\in S^{N-1,1}_\mathbb R[/math], as desired.

(3) [math]S^{N-1}_{\mathbb R,*}\cap\bar{S}^{N-1}_\mathbb R[/math] is obtained from [math]\bar{S}^{N-1}_\mathbb R[/math] by imposing to the standard coordinates the half-commutation relations [math]abc=cba[/math]. On the other hand, we know from [math]\bar{S}^{N-1}_\mathbb R\subset \bar{S}^{N-1}_{\mathbb R,*}[/math] that the standard coordinates on [math]\bar{S}^{N-1}_\mathbb R[/math] satisfy [math]abc=-cba[/math] for [math]a,b,c[/math] distinct, and [math]abc=cba[/math] otherwise. Thus, the relations brought by intersecting with [math]S^{N-1}_{\mathbb R,*}[/math] reduce to the relations [math]abc=0[/math] for [math]a,b,c[/math] distinct, and so we are led to the sphere [math]\bar{S}^{N-1,1}_\mathbb R[/math].

(4) [math]S^{N-1}_{\mathbb R,*}\cap\bar{S}^{N-1}_{\mathbb R,*}[/math] is obtained from [math]\bar{S}^{N-1}_{\mathbb R,*}[/math] by imposing the relations [math]abc=-cba[/math] for [math]a,b,c[/math] distinct, and [math]abc=cba[/math] otherwise. Since we know that [math]abc=cba[/math] for any [math]a,b,c[/math], the extra relations reduce to [math]abc=0[/math] for [math]a,b,c[/math] distinct, and so we are led to [math]S^{N-1,1}_{\mathbb R,*}[/math].

■

Summarizing, whether we want it or not, when talking about intersections between twisted and untwisted geometries, we are led into polygonal spheres, and into non-smooth objects in general. In view of this, and also in connection with general axiomatization questions, let us find now a suitable axiomatic framework for the 9 spheres in Theorem 13.17. We have the following definition, once again from , which is based on the signature function [math]\varepsilon:P_{even}\to\{\pm1\}[/math] constructed in chapter 11:

Definition

Given variables [math]x_1,\ldots,x_N[/math], any permutation [math]\sigma\in S_k[/math] produces two collections of relations between these variables, as follows:

• Untwisted relations, namely, for any [math]i_1,\ldots,i_k[/math]:
  [[math]] x_{i_1}\ldots x_{i_k}=x_{i_{\sigma(1)}}\ldots x_{i_{\sigma(k)}} [[/math]]
• Twisted relations, namely, for any [math]i_1,\ldots,i_k[/math]:
  [[math]] x_{i_1}\ldots x_{i_k}=\varepsilon\left(\ker\begin{pmatrix}i_1&\ldots&i_k\\ i_{\sigma(1)}&\ldots&i_{\sigma(k)}\end{pmatrix}\right)x_{i_{\sigma(1)}}\ldots x_{i_{\sigma(k)}} [[/math]]

The untwisted relations are denoted [math]\mathcal R_\sigma[/math], and the twisted ones are denoted [math]\bar{\mathcal R}_\sigma[/math].

Observe that the untwisted relations [math]\mathcal R_\sigma[/math] are trivially satisfied for the standard coordinates on [math]S^{N-1}_\mathbb R[/math], for any permutation [math]\sigma\in S_k[/math]. A twisted analogue of this fact holds, in the sense that the standard coordinates on [math]\bar{S}^{N-1}_\mathbb R[/math] satisfy the relations [math]\bar{\mathcal R}_\sigma[/math], for any [math]\sigma\in S_k[/math]. Indeed, by using the anticommutation relations between the distinct coordinates of these latter spheres, we must have a formula of the following type:

[[math]] x_{i_1}\ldots x_{i_k}=\pm x_{i_{\sigma(1)}}\ldots x_{i_{\sigma(k)}} [[/math]]

But the sign [math]\pm[/math] obtained in this way is precisely the one given above, namely:

[[math]] \pm=\varepsilon\left(\ker\begin{pmatrix}i_1&\ldots&i_k\\ i_{\sigma(1)}&\ldots&i_{\sigma(k)}\end{pmatrix}\right) [[/math]]

We have now all the needed ingredients for axiomatizing the various spheres appearing so far, namely the twisted and untwisted ones, and their intersections:

Definition

We have [math]3[/math] types of quantum spheres [math]S\subset S^{N-1}_{\mathbb R,+}[/math], as follows:

• Monomial, namely [math]\dot{S}^{N-1}_{\mathbb R,E}[/math], with [math]E\subset S_\infty[/math], obtained via the following relations:
  [[math]] \left\{\dot{\mathcal R}_\sigma\Big|\sigma\in E\right\} [[/math]]
• Mixed monomial, which appear as intersections as follows, with [math]E,F\subset S_\infty[/math]:
  [[math]] S^{N-1}_{\mathbb R,E,F}=S^{N-1}_{\mathbb R,E}\cap\bar{S}^{N-1}_{\mathbb R,F} [[/math]]
• Polygonal, which are again intersections, with [math]E,F\subset S_\infty[/math], and [math]d\in\{1,\ldots,N\}[/math]:
  [[math]] S^{N-1,d-1}_{\mathbb R,E,F}=S^{N-1}_{\mathbb R,E,F}\cap S^{N-1,d-1}_{\mathbb R,+} [[/math]]

With the above notions, we cover all spheres appearing so far. More precisely, the 5 basic spheres in are monomial, the 9 spheres in Theorem 13.17 are mixed monomial, and the polygonal sphere formalism covers all the examples constructed so far.

Observe that the set of mixed monomial spheres is closed under intersections. The same holds for the set of polygonal spheres, because we have the following formula:

[[math]] S^{N-1,d-1}_{\mathbb R,E,F}\cap S^{N-1,d'-1}_{\mathbb R,E',F'}=S^{N-1,min(d,d')-1}_{\mathbb R,E\cup E',F\cup F'} [[/math]]

Let us try now to understand the structure of the various types of spheres, by using the real sphere technology developed before. We call a group of permutations [math]G\subset S_\infty[/math] filtered if, with [math]G_k=G\cap S_k[/math], we have [math]G_k\times G_l\subset G_{k+l}[/math], for any [math]k,l[/math]. We have:

Proposition

The various spheres can be parametrized by groups, as follows:

• Monomial case: [math]\dot{S}^{N-1}_{\mathbb R,G}[/math], with [math]G\subset S_\infty[/math] filtered group.
• Mixed monomial case: [math]S^{N-1}_{\mathbb R,G,H}[/math], with [math]G,H\subset S_\infty[/math] filtered groups.
• Polygonal case: [math]S^{N-1,d-1}_{\mathbb R,G,H}[/math], with [math]G,H\subset S_\infty[/math] filtered groups, and [math]d\in\{1,\ldots,N\}[/math].

Show Proof

This basically follows from the theory developed before, as follows:

(1) As explained before, in order to prove this assertion, for a monomial sphere [math]S=\dot{S}_{\mathbb R,E}[/math], we can take [math]G\subset S_\infty[/math] to be the set of permutations [math]\sigma\in S_\infty[/math] having the property that the relations [math]\dot{\mathcal R}_\sigma[/math] hold for the standard coordinates of [math]S[/math]. We have then [math]E\subset G[/math], we have as well [math]S=\dot{S}^{N-1}_{\mathbb R,G}[/math], and the fact that [math]G[/math] is a filtered group is clear as well.

(2) This follows from (1), by taking intersections.

(3) Once again this follows from (1), by taking intersections.

■

The idea in what follows will be that of writing the 9 main polygonal spheres as in Proposition 13.20 (2), as to reach to a “standard parametrization” for our spheres. We recall that the permutations [math]\sigma\in S_\infty[/math] having the property that when labelling clockwise their legs [math]\circ\bullet\circ\bullet\ldots[/math], and string joins a white leg to a black leg, form a filtered group, denoted [math]S_\infty^*\subset S_\infty[/math]. This group comes from the general half-liberation considerations from chapter 9, and its algebraic structure is very simple, as follows:

[[math]] S_{2n}^*\simeq S_n\times S_n\quad,\quad S_{2n+1}^*\simeq S_n\times S_{n+1} [[/math]]

Let us formulate as well the following definition:

Definition

We call a mixed monomial sphere parametrization

[[math]] S=S^{N-1}_{\mathbb R,G,H} [[/math]]

standard when both filtered groups [math]G,H\subset S_\infty[/math] are chosen to be maximal.

In this case, Proposition 13.20 and its proof tell us that [math]G,H[/math] encode all the monomial relations which hold in [math]S[/math]. With these conventions, we have the following result from , , extending some previous findings from above, regarding the untwisted spheres:

Theorem

The standard parametrization of the [math]9[/math] main spheres is

[[math]] \xymatrix@R=11.5mm@C=11.5mm{ S_\infty\ar@{.}[d]&S_\infty^*\ar@{.}[d]&\{1\}\ar@{.}[d]&G/H\\ S^{N-1}_\mathbb R\ar[r]&S^{N-1}_{\mathbb R,*}\ar[r]&S^{N-1}_{\mathbb R,+}&\{1\}\ar@{.}[l]\\ S^{N-1,1}_\mathbb R\ar[r]\ar[u]&S^{N-1,1}_{\mathbb R,*}\ar[r]\ar[u]&\bar{S}^{N-1}_{\mathbb R,*}\ar[u]&S_\infty^*\ar@{.}[l]\\ S^{N-1,0}_\mathbb R\ar[r]\ar[u]&\bar{S}^{N-1,1}_\mathbb R\ar[r]\ar[u]&\bar{S}^{N-1}_\mathbb R\ar[u]&S_\infty\ar@{.}[l]} [[/math]]

so these spheres come from the [math]3\times 3=9[/math] pairs of groups among [math]\{1\}\subset S_\infty^*\subset S_\infty[/math].

Show Proof

The fact that we have parametrizations as above is known to hold for the 5 untwisted and twisted spheres. For the remaining 4 spheres the result follows by intersecting, by using the following formula, valid for any [math]E,F\subset S_\infty[/math]:

[[math]] S^{N-1}_{\mathbb R,E,F}\cap S^{N-1}_{\mathbb R,E',F'}=S^{N-1}_{\mathbb R,E\cup E',F\cup F'} [[/math]]

In order to prove now that the parametrizations are standard, we must compute the following two filtered groups, and show that we get the groups in the statement:

[[math]] G=\left\{\sigma\in S_\infty\Big|{\rm the\ relations\ }\mathcal R_\sigma\ {\rm hold\ over\ }S\right\} [[/math]]

[[math]] H=\left\{\sigma\in S_\infty\Big|{\rm the\ relations\ }\bar{\mathcal R}_\sigma\ {\rm hold\ over\ }S\right\} [[/math]]

As a first observation, by using the various inclusions between spheres, we just have to compute [math]G[/math] for the spheres on the bottom, and [math]H[/math] for the spheres on the left:

[[math]] X=S^{N-1,0}_\mathbb R,\bar{S}^{N-1,1}_\mathbb R,\bar{S}^{N-1}_\mathbb R\implies G=S_\infty,S_\infty^*,\{1\} [[/math]]

[[math]] X=S^{N-1,0}_\mathbb R,S^{N-1,1}_\mathbb R,S^{N-1}_\mathbb R\implies H=S_\infty,S_\infty^*,\{1\} [[/math]]

The results for [math]S^{N-1,0}_\mathbb R[/math] being clear, we are left with computing the remaining 4 groups, for the spheres [math]S^{N-1}_\mathbb R,\bar{S}^{N-1}_\mathbb R,S^{N-1,1}_\mathbb R,\bar{S}^{N-1,1}_\mathbb R[/math]. The proof here goes as follows:

(1) [math]S^{N-1}_\mathbb R[/math]. According to the definition of [math]H=(H_k)[/math], we have:

[[math]] \begin{eqnarray*} H_k &=&\left\{\sigma\in S_k\Big|x_{i_1}\ldots x_{i_k}=\varepsilon\left( \ker\begin{pmatrix}i_1&\ldots&i_k\\ i_{\sigma(1)}&\ldots&i_{\sigma(k)} \end{pmatrix}\right) x_{i_{\sigma(1)}}\ldots x_{i_{\sigma(k)}},\forall i_1,\ldots,i_k\right\}\\ &=&\left\{\sigma\in S_k\Big| \varepsilon\left( \ker\begin{pmatrix}i_1&\ldots&i_k\\ i_{\sigma(1)}&\ldots&i_{\sigma(k)} \end{pmatrix}\right) =1,\forall i_1,\ldots,i_k\right\}\\ &=&\left\{\sigma\in S_k\Big|\varepsilon(\tau)=1,\forall\tau\leq\sigma\right\}\end{eqnarray*} [[/math]]

Now observe that for any permutation [math]\sigma\in S_k,\sigma\neq1_k[/math], we can always find a partition [math]\tau\leq\sigma[/math] satisfying the following condition:

[[math]] \varepsilon(\tau)=-1 [[/math]]

We deduce that we have [math]H_k=\{1_k\}[/math], and so [math]H=\{1\}[/math], as desired.

(2) [math]\bar{S}^{N-1}_\mathbb R[/math]. The proof of [math]G=\{1\}[/math] here is similar to the proof of [math]H=\{1\}[/math] in (1) above, by using the same combinatorial ingredient at the end.

(3) [math]S^{N-1,1}_\mathbb R[/math]. By definition of [math]H=(H_k)[/math], a permutation [math]\sigma\in S_k[/math] belongs to [math]H_k[/math] when the following condition is satisfied, for any choice of the indices [math]i_1,\ldots,i_k[/math]:

[[math]] x_{i_1}\ldots x_{i_k}=\varepsilon\left(\ker\begin{pmatrix}i_1&\ldots&i_k\\ i_{\sigma(1)}&\ldots&i_{\sigma(k)}\end{pmatrix}\right)x_{i_{\sigma(1)}}\ldots x_{i_{\sigma(k)}} [[/math]]

We have three cases here, as follows:

-- When [math]|\ker i|=1[/math] this formula reads [math]x_r^k=x_r^k[/math], which is true.

-- When [math]|\ker i|\geq3[/math] this formula is automatically satisfied as well, because by using the relations [math]ab=ba[/math], and [math]abc=0[/math] for [math]a,b,c[/math] distinct, which both hold over [math]S^{N-1,1}_\mathbb R[/math], this formula reduces to [math]0=0[/math].

-- Thus, we are left with studying the case [math]|\ker i|=2[/math]. Here the quantities on the left [math]x_{i_1}\ldots x_{i_k}[/math] will not vanish, so the sign on the right must be 1, and we therefore have:

[[math]] H_k=\left\{\sigma\in S_k\Big|\varepsilon(\tau)=1,\forall\tau\leq\sigma,|\tau|=2\right\} [[/math]]

Now by coloring the legs of [math]\sigma[/math] clockwise [math]\circ\bullet\circ\bullet\ldots[/math], the above condition is satisfied when each string of [math]\sigma[/math] joins a white leg to a black leg. Thus [math]H_k=S_k^*[/math], as desired.

(4) [math]\bar{S}^{N-1,1}_\mathbb R[/math]. The proof of [math]G=S_\infty^*[/math] here is similar to the proof of [math]H=S_\infty^*[/math] in (3) above, by using the same combinatorial ingredient at the end.

■

We will be back to the polygonal spheres in the next chapter, with a better axiomatization, and with a study of the associated quantum groups as well.

General references

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