guide:Ae227bcb0f

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

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Getting back now to our general projective geometry program, we would like to have axiomatization and classification results for such quadruplets. In order to do this, following , we can axiomatize our various projective spaces, as follows:

Definition

A monomial projective space is a closed subset [math]P\subset P^{N-1}_+[/math] obtained via relations of type

[[math]] p_{i_1i_2}\ldots p_{i_{k-1}i_k}=p_{i_{\sigma(1)}i_{\sigma(2)}}\ldots p_{i_{\sigma(k-1)}i_{\sigma(k)}},\ \forall (i_1,\ldots,i_k)\in\{1,\ldots,N\}^k [[/math]]

with [math]\sigma[/math] ranging over a certain subset of [math]\bigcup_{k\in2\mathbb N}S_k[/math], which is stable under [math]\sigma\to|\sigma|[/math].

Observe the similarity with the corresponding monomiality notion for the spheres, from chapter 13. The only subtlety in the projective case is the stability under the operation [math]\sigma\to|\sigma|[/math], which in practice means that if the above relation associated to [math]\sigma[/math] holds, then the following relation, associated to [math]|\sigma|[/math], must hold as well:

[[math]] p_{i_0i_1}\ldots p_{i_ki_{k+1}}=p_{i_0i_{\sigma(1)}}p_{i_{\sigma(2)}i_{\sigma(3)}}\ldots p_{i_{\sigma(k-2)}i_{\sigma(k-1)}}p_{i_{\sigma(k)}i_{k+1}} [[/math]]

As an illustration, the basic projective spaces are all monomial:

Proposition

The [math]3[/math] projective spaces are all monomial, with the permutations

[[math]] \xymatrix@R=10mm@C=8mm{\circ\ar@{-}[dr]&\circ\ar@{-}[dl]\\\circ&\circ}\qquad\ \qquad\ \qquad \item[a]ymatrix@R=10mm@C=3mm{\circ\ar@{-}[drr]&\circ\ar@{-}[drr]&\circ\ar@{-}[dll]&\circ\ar@{-}[dll]\\\circ&\circ&\circ&\circ} [[/math]]

producing respectively the spaces [math]P^{N-1}_\mathbb R,P^{N-1}_\mathbb C[/math], and with no relation needed for [math]P^{N-1}_+[/math].

Show Proof

We must divide the algebra [math]C(P^{N-1}_+)[/math] by the relations associated to the diagrams in the statement, as well as those associated to their shifted versions, given by:

[[math]] \xymatrix@R=10mm@C=3mm{\circ\ar@{-}[d]&\circ\ar@{-}[dr]&\circ\ar@{-}[dl]&\circ\ar@{-}[d]\\\circ&\circ&\circ&\circ} \qquad\ \qquad\ \qquad \item[a]ymatrix@R=10mm@C=3mm{\circ\ar@{-}[d]&\circ\ar@{-}[drr]&\circ\ar@{-}[drr]&\circ\ar@{-}[dll]&\circ\ar@{-}[dll]&\circ\ar@{-}[d]\\\circ&\circ&\circ&\circ&\circ&\circ} [[/math]]

(1) The basic crossing, and its shifted version, produce the following relations:

[[math]] p_{ab}=p_{ba} [[/math]]

[[math]] p_{ab}p_{cd}=p_{ac}p_{bd} [[/math]]

Now by using these relations several times, we obtain the following formula:

[[math]] p_{ab}p_{cd} =p_{ac}p_{bd} =p_{ca}p_{db} =p_{cd}p_{ab} [[/math]]

Thus, the space produced by the basic crossing is classical, [math]P\subset P^{N-1}_\mathbb C[/math]. By using one more time the relations [math]p_{ab}=p_{ba}[/math] we conclude that we have [math]P=P^{N-1}_\mathbb R[/math], as claimed.

(2) The fattened crossing, and its shifted version, produce the following relations:

[[math]] p_{ab}p_{cd}=p_{cd}p_{ab} [[/math]]

[[math]] p_{ab}p_{cd}p_{ef}=p_{ad}p_{eb}p_{cf} [[/math]]

The first relations tell us that the projective space must be classical, [math]P\subset P^{N-1}_\mathbb C[/math]. Now observe that with [math]p_{ij}=z_i\bar{z}_j[/math], the second relations read:

[[math]] z_a\bar{z}_bz_c\bar{z}_dz_e\bar{z}_f=z_a\bar{z}_dz_e\bar{z}_bz_c\bar{z}_f [[/math]]

Since these relations are automatic, we have [math]P=P^{N-1}_\mathbb C[/math], and we are done.

■

Following , we can now formulate our classification result, as follows:

Theorem

The basic projective spaces, namely

[[math]] P^{N-1}_\mathbb R\subset P^{N-1}_\mathbb C\subset P^{N-1}_+ [[/math]]

are the only monomial ones.

Show Proof

We follow the proof from the affine case. Let [math]\mathcal R_\sigma[/math] be the collection of relations associated to a permutation [math]\sigma\in S_k[/math] with [math]k\in 2\mathbb N[/math], as in Definition 15.11. We fix a monomial projective space [math]P\subset P^{N-1}_+[/math], and we associate to it subsets [math]G_k\subset S_k[/math], as follows:

[[math]] G_k=\begin{cases} \{\sigma\in S_k|\mathcal R_\sigma\ {\rm hold\ over\ }P\}&(k\ {\rm even})\\ \{\sigma\in S_k|\mathcal R_{|\sigma}\ {\rm hold\ over\ }P\}&(k\ {\rm odd}) \end{cases} [[/math]]

As in the affine case, we obtain in this way a filtered group [math]G=(G_k)[/math], which is stable under removing outer strings, and under removing neighboring strings. Thus the computations in chapter 13 apply, and show that we have only 3 possible situations, corresponding to the 3 projective spaces in Proposition 15.12.

■

Let us discuss now similar results for the projective quantum groups. Given a closed subgroup [math]G\subset O_N^+[/math], its projective version [math]G\to PG[/math] is by definition given by the fact that [math]C(PG)\subset C(G)[/math] is the subalgebra generated by the following variables:

[[math]] w_{ij,ab}=u_{ia}u_{jb} [[/math]]

In the classical case we recover in this way the usual projective version:

[[math]] PG=G/(G\cap\mathbb Z_2^N) [[/math]]

We have the following key result, from :

Theorem

The quantum group [math]O_N^*[/math] is the unique intermediate easy quantum group [math]O_N\subset G\subset O_N^+[/math]. Moreover, in the non-easy case, the following happen:

• The group inclusion [math]\mathbb TO_N\subset U_N[/math] is maximal.
• The group inclusion [math]PO_N\subset PU_N[/math] is maximal.
• The quantum group inclusion [math]O_N\subset O_N^*[/math] is maximal.

Show Proof

This is something discussed in chapters 9-10, the idea being that the first assertion comes by classifying the categories of pairings, and then:

(1) This can be obtained by using standard Lie group methods.

(2) This follows from (1), by taking projective versions.

(3) This follows from (2), via standard algebraic lifting results.

■

Our claim now is that, under suitable assumptions, [math]PU_N[/math] is the only intermediate object [math]PO_N\subset G\subset PO_N^+[/math]. In order to formulate a precise statement here, we first recall the following notion, from , that we have already heavily used in this book:

Definition

A collection of sets [math]D=\bigsqcup_{k,l}D(k,l)[/math] with

[[math]] D(k,l)\subset P(k,l) [[/math]]

is called a category of partitions when it has the following properties:

• Stability under the horizontal concatenation, [math](\pi,\sigma)\to[\pi\sigma][/math].
• Stability under vertical concatenation [math](\pi,\sigma)\to[^\sigma_\pi][/math], with matching middle symbols.
• Stability under the upside-down turning [math]*[/math], with switching of colors, [math]\circ\leftrightarrow\bullet[/math].
• Each set [math]P(k,k)[/math] contains the identity partition [math]||\ldots||[/math].
• The sets [math]P(\emptyset,\circ\bullet)[/math] and [math]P(\emptyset,\bullet\circ)[/math] both contain the semicircle [math]\cap[/math].

The above definition is something inspired from the axioms of Tannakian categories, and going hand in hand with it is the following definition, also from :

Definition

An intermediate compact quantum group

[[math]] O_N\subset G\subset O_N^+ [[/math]]

is called easy when the corresponding Tannakian category

[[math]] span(NC_2(k,l))\subset Hom(u^{\otimes k},u^{\otimes l})\subset span(P_2(k,l)) [[/math]]

comes via the following formula, using the standard [math]\pi\to T_\pi[/math] construction,

[[math]] Hom(u^{\otimes k},u^{\otimes l})=span(D(k,l)) [[/math]]

from a certain collection of sets of pairings [math]D=(D(k,l))[/math].

As explained in , by “saturating” the sets [math]D(k,l)[/math], we can assume that the collection [math]D=(D(k,l))[/math] is a category of pairings, in the sense that it is stable under vertical and horizontal concatenation, upside-down turning, and contains the semicircle.

In the projective case now, following , let us formulate:

Definition

A projective category of pairings is a collection of subsets

[[math]] NC_2(2k,2l)\subset E(k,l)\subset P_2(2k,2l) [[/math]]

stable under the usual categorical operations, and satisfying [math]\sigma\in E\implies |\sigma|\in E[/math].

As basic examples here, we have the following projective categories of pairings, where [math]P_2^*[/math] is the category of matching pairings:

[[math]] NC_2\subset P_2^*\subset P_2 [[/math]]

This follows indeed from definitions. Now with the above notion in hand, we can formulate the following projective analogue of the notion of easiness:

Definition

An intermediate compact quantum group

[[math]] PO_N\subset H\subset PO_N^+ [[/math]]

is called projectively easy when its Tannakian category

[[math]] span(NC_2(2k,2l))\subset Hom(v^{\otimes k},v^{\otimes l})\subset span(P_2(2k,2l)) [[/math]]

comes via via the following formula, using the standard [math]\pi\to T_\pi[/math] construction,

[[math]] Hom(v^{\otimes k},v^{\otimes l})=span(E(k,l)) [[/math]]

for a certain projective category of pairings [math]E=(E(k,l))[/math].

Thus, we have a projective notion of easiness. Observe that, given an easy quantum group [math]O_N\subset G\subset O_N^+[/math], its projective version [math]PO_N\subset PG\subset PO_N^+[/math] is projectively easy in our sense. In particular the basic projective quantum groups [math]PO_N\subset PU_N\subset PO_N^+[/math] are all projectively easy in our sense, coming from the categories [math]NC_2\subset P_2^*\subset P_2[/math].

We have in fact the following general result, from :

Theorem

We have a bijective correspondence between the affine and projective categories of partitions, given by the operation

[[math]] G\to PG [[/math]]

at the level of the corresponding affine and projective easy quantum groups.

Show Proof

The construction of correspondence [math]D\to E[/math] is clear, simply by setting:

[[math]] E(k,l)=D(2k,2l) [[/math]]

Indeed, due to the axioms in Definition 15.15, the conditions in Definition 15.17 are satisfied. Conversely, given [math]E=(E(k,l))[/math] as in Definition 15.17, we can set:

[[math]] D(k,l)=\begin{cases} E(k,l)&(k,l\ {\rm even})\\ \{\sigma:|\sigma\in E(k+1,l+1)\}&(k,l\ {\rm odd}) \end{cases} [[/math]]

Our claim is that [math]D=(D(k,l))[/math] is a category of partitions. Indeed:

(1) The composition action is clear. Indeed, when looking at the numbers of legs involved, in the even case this is clear, and in the odd case, this follows from:

[[math]] \begin{eqnarray*} |\sigma,|\sigma'\in E &\implies&|^\sigma_\tau\in E\\ &\implies&{\ }^\sigma_\tau\in D \end{eqnarray*} [[/math]]

(2) For the tensor product axiom, we have 4 cases to be investigated, depending on the parity of the number of legs of [math]\sigma,\tau[/math], as follows:

-- The even/even case is clear.

-- The odd/even case follows from the following computation:

[[math]] \begin{eqnarray*} |\sigma,\tau\in E &\implies&|\sigma\tau\in E\\ &\implies&\sigma\tau\in D \end{eqnarray*} [[/math]]

-- Regarding now the even/odd case, this can be solved as follows:

[[math]] \begin{eqnarray*} \sigma,|\tau\in E &\implies&|\sigma|,|\tau\in E\\ &\implies&|\sigma||\tau\in E\\ &\implies&|\sigma\tau\in E\\ &\implies&\sigma\tau\in D \end{eqnarray*} [[/math]]

-- As for the remaining odd/odd case, here the computation is as follows:

[[math]] \begin{eqnarray*} |\sigma,|\tau\in E &\implies&||\sigma|,|\tau\in E\\ &\implies&||\sigma||\tau\in E\\ &\implies&\sigma\tau\in E\\ &\implies&\sigma\tau\in D \end{eqnarray*} [[/math]]

(3) Finally, the conjugation axiom is clear from definitions. It is also clear that both compositions [math]D\to E\to D[/math] and [math]E\to D\to E[/math] are the identities, as claimed. As for the quantum group assertion, this is clear as well from definitions.

■

Now back to uniqueness issues, we have here the following result, also from :

Theorem

We have the following results:

• [math]O_N^*[/math] is the only intermediate easy quantum group, as follows:
  [[math]] O_N\subset G\subset O_N^+ [[/math]]
• [math]PU_N[/math] is the only intermediate projectively easy quantum group, as follows:
  [[math]] PO_N\subset G\subset PO_N^+ [[/math]]

Show Proof

The idea here is as follows:

(1) The assertion regarding [math]O_N\subset O_N^*\subset O_N^+[/math] is from , and this is something that we already know, explained in chapter 9.

(2) The assertion regarding [math]PO_N\subset PU_N\subset PO_N^+[/math] follows from the classification result in (1), and from the duality in Theorem 15.19.

■

Summarizing, we have analogues of the various affine classification results, with the remark that everything becomes simpler in the projective setting.

General references

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