15c. Projective geometry

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We have so far projective analogues of the various affine classification results. In view of this, our next goal will be that of finding projective versions of the quantum isometry group results that we have in the affine setting. We use the following action formalism, which is quite similar to the affine action formalism introduced in chapter 3:

Definition

Consider a closed subgroup of the free orthogonal group, [math]G\subset O_N^+[/math], and a closed subset of the free real sphere, [math]X\subset S^{N-1}_{\mathbb R,+}[/math].

• We write [math]G\curvearrowright X[/math] when we have a morphism of [math]C^*[/math]-algebras, as follows:
  [[math]] \Phi:C(X)\to C(X)\otimes C(G) [[/math]]
  [[math]] \Phi(z_i)=\sum_az_a\otimes u_{ai} [[/math]]
• We write [math]PG\curvearrowright PX[/math] when we have a morphism of [math]C^*[/math]-algebras, as follows:
  [[math]] \Phi:C(PX)\to C(PX)\otimes C(PG) [[/math]]
  [[math]] \Phi(z_iz_j)=\sum_a z_az_b\otimes u_{ai}u_{bj} [[/math]]

Observe that the above morphisms [math]\Phi[/math], if they exist, are automatically coaction maps. Observe also that an affine action [math]G\curvearrowright X[/math] produces a projective action [math]PG\curvearrowright PX[/math]. Let us also mention that given an algebraic subset [math]X\subset S^{N-1}_{\mathbb R,+}[/math], it is routine to prove that there exist indeed universal quantum groups [math]G\subset O_N^+[/math] acting as (1), and as in (2). We have the following result, from and related papers, with respect to the above notions:

Theorem

The quantum isometry groups of basic spheres and projective spaces,

[[math]] \xymatrix@R=15mm@C=15mm{ S^{N-1}_\mathbb R\ar[r]\ar[d]&S^{N-1}_{\mathbb R,*}\ar[r]\ar[d]&S^{N-1}_{\mathbb R,+}\ar[d]\\ P^{N-1}_\mathbb R\ar[r]&P^{N-1}_\mathbb C\ar[r]&P^{N-1}_+} [[/math]]

are the following affine and projective quantum groups,

[[math]] \xymatrix@R=15mm@C=15mm{ O_N\ar[r]\ar[d]&O_N^*\ar[r]\ar[d]&O_N^+\ar[d]\\ PO_N\ar[r]&PU_N\ar[r]&PO_N^+} [[/math]]

with respect to the affine and projective action notions introduced above.

Show Proof

The fact that the 3 quantum groups on top act affinely on the corresponding 3 spheres is known since , and is elementary, explained before. By restriction, the 3 quantum groups on the bottom follow to act on the corresponding 3 projective spaces. We must prove now that all these actions are universal. At right there is nothing to prove, so we are left with studying the actions on [math]S^{N-1}_\mathbb R,S^{N-1}_{\mathbb R,*}[/math] and on [math]P^{N-1}_\mathbb R,P^{N-1}_\mathbb C[/math].

\underline{[math]P^{N-1}_\mathbb R[/math]}. Consider the following projective coordinates:

[[math]] p_{ij}=z_iz_j\quad,\quad w_{ij,ab}=u_{ai}u_{bj} [[/math]]

In terms of these projective coordinates, the coaction map is given by:

[[math]] \Phi(p_{ij})=\sum_{ab}p_{ab}\otimes w_{ij,ab} [[/math]]

Thus, we have the following formulae:

[[math]] \begin{eqnarray*} \Phi(p_{ij})&=&\sum_{a \lt b}p_{ab}\otimes (w_{ij,ab}+w_{ij,ba})+\sum_ap_{aa}\otimes w_{ij,aa}\\ \Phi(p_{ji})&=&\sum_{a \lt b}p_{ab}\otimes (w_{ji,ab}+w_{ji,ba})+\sum_ap_{aa}\otimes w_{ji,aa} \end{eqnarray*} [[/math]]

By comparing these two formulae, and then by using the linear independence of the variables [math]p_{ab}=z_az_b[/math] for [math]a\leq b[/math], we conclude that we must have:

[[math]] w_{ij,ab}+w_{ij,ba}=w_{ji,ab}+w_{ji,ba} [[/math]]

Let us apply now the antipode to this formula. For this purpose, observe that:

[[math]] \begin{eqnarray*} S(w_{ij,ab}) &=&S(u_{ai}u_{bj})\\ &=&S(u_{bj})S(u_{ai})\\ &=&u_{jb}u_{ia}\\ &=&w_{ba,ji} \end{eqnarray*} [[/math]]

Thus by applying the antipode we obtain:

[[math]] w_{ba,ji}+w_{ab,ji}=w_{ba,ij}+w_{ab,ij} [[/math]]

By relabelling, we obtain the following formula:

[[math]] w_{ji,ba}+w_{ij,ba}=w_{ji,ab}+w_{ij,ab} [[/math]]

Now by comparing with the original relation, we obtain:

[[math]] w_{ij,ab}=w_{ji,ba} [[/math]]

But, with [math]w_{ij,ab}=u_{ai}u_{bj}[/math], this formula reads:

[[math]] u_{ai}u_{bj}=u_{bj}u_{ai} [[/math]]

Thus [math]G\subset O_N[/math], and it follows that we have [math]PG\subset PO_N[/math], as claimed.

\underline{[math]P^{N-1}_\mathbb C[/math]}. Consider a coaction map, written as follows, with [math]p_{ab}=z_a\bar{z}_b[/math]:

[[math]] \Phi(p_{ij})=\sum_{ab}p_{ab}\otimes u_{ai}u_{bj} [[/math]]

The idea here will be that of using the following formula:

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

We have the following formulae:

[[math]] \begin{eqnarray*} \Phi(p_{ij}p_{kl})&=&\sum_{abcd}p_{ab}p_{cd}\otimes u_{ai}u_{bj}u_{ck}u_{dl}\\ \Phi(p_{il}p_{kj})&=&\sum_{abcd}p_{ad}p_{cb}\otimes u_{ai}u_{dl}u_{ck}u_{bj} \end{eqnarray*} [[/math]]

The terms at left being equal, and the last terms at right being equal too, we deduce that, with [math][a,b,c]=abc-cba[/math], we must have the following formula:

[[math]] \sum_{abcd}u_{ai}[u_{bj},u_{ck},u_{dl}]\otimes p_{ab}p_{cd}=0 [[/math]]

Now since the quantities [math]p_{ab}p_{cd}=z_a\bar{z}_bz_c\bar{z}_d[/math] at right depend only on the numbers [math]|\{a,c\}|,|\{b,d\}|\in\{1,2\}[/math], and this dependence produces the only possible linear relations between the variables [math]p_{ab}p_{cd}[/math], we are led to [math]2\times2=4[/math] equations, as follows:

(1) [math]u_{ai}[u_{bj},u_{ak},u_{bl}]=0[/math], [math]\forall a,b[/math].

(2) [math]u_{ai}[u_{bj},u_{ak},u_{dl}]+u_{ai}[u_{dj},u_{ak},u_{bl}]=0[/math], [math]\forall a[/math], [math]\forall b\neq d[/math].

(3) [math]u_{ai}[u_{bj},u_{ck},u_{bl}]+u_{ci}[u_{bj},u_{ak},u_{bl}]=0[/math], [math]\forall a\neq c[/math], [math]\forall b[/math].

(4) [math]u_{ai}[u_{bj},u_{ck},u_{dl}]+u_{ai}[u_{dj},u_{ck},u_{bl}]+u_{ci}[u_{bj},u_{ak},u_{dl}]+u_{ci}[u_{dj},u_{ak},u_{bl}]=0[/math], [math]\forall a\neq c,b\neq d[/math].

We will need in fact only the first two formulae. Since (1) corresponds to (2) at [math]b=d[/math], we conclude that (1,2) are equivalent to (2), with no restriction on the indices. By multiplying now this formula to the left by [math]u_{ai}[/math], and then summing over [math]i[/math], we obtain:

[[math]] [u_{bj},u_{ak},u_{dl}]+[u_{dj},u_{ak},u_{bl}]=0 [[/math]]

We use now the antipode/relabel trick from . By applying the antipode we obtain:

[[math]] [u_{ld},u_{ka},u_{jb}]+[u_{lb},u_{ka},u_{jd}]=0 [[/math]]

By relabelling we obtain the following formula:

[[math]] [u_{dl},u_{ak},u_{bj}]+[u_{dj},u_{ak},u_{bl}]=0 [[/math]]

Now by comparing with the original relation, we obtain:

[[math]] [u_{bj},u_{ak},u_{dl}]=[u_{dj},u_{ak},u_{bl}]=0 [[/math]]

Thus [math]G\subset O_N^*[/math], and it follows that we have [math]PG\subset PU_N[/math], as desired.

■

The above results can be probably improved. As an example, let us say that a closed subgroup [math]G\subset U_N^+[/math] acts projectively on [math]PX[/math] when we have a coaction map as follows:

[[math]] \Phi(z_iz_j)=\sum_{ab}z_az_b\otimes u_{ai}u_{bj}^* [[/math]]

The above proof can be adapted, by putting [math]*[/math] signs where needed, and Theorem 15.22 still holds, in this setting. However, establishing general universality results, involving arbitrary subgroups [math]H\subset PO_N^+[/math], looks like a quite non-trivial question.

Let us discuss now the axiomatization question for the projective quadruplets of type [math](P,PT,PU,PK)[/math]. We recall that we first have a classical real quadruplet, as follows:

[[math]] \xymatrix@R=50pt@C=50pt{ P^{N-1}_\mathbb R\ar@{-}[r]\ar@{-}[d]\ar@{-}[dr]&PT_N\ar@{-}[l]\ar@{-}[d]\ar@{-}[dl]\\ PO_N\ar@{-}[u]\ar@{-}[ur]\ar@{-}[r]&PH_N\ar@{-}[l]\ar@{-}[ul]\ar@{-}[u]} [[/math]]

We have then a classical complex quadruplet, which can be thought of as well as being a real half-classical quadruplet, which is as follows:

[[math]] \xymatrix@R=50pt@C=50pt{ P^{N-1}_\mathbb C\ar@{-}[r]\ar@{-}[d]\ar@{-}[dr]&P\mathbb T_N\ar@{-}[l]\ar@{-}[d]\ar@{-}[dl]\\ PU_N\ar@{-}[u]\ar@{-}[ur]\ar@{-}[r]&PK_N\ar@{-}[l]\ar@{-}[ul]\ar@{-}[u]} [[/math]]

Finally, we have a free quadruplet, which can be thought of as being the same time real and complex, which is as follows:

[[math]] \xymatrix@R=50pt@C=50pt{ P^{N-1}_+\ar@{-}[r]\ar@{-}[d]\ar@{-}[dr]&PT_N^+\ar@{-}[l]\ar@{-}[d]\ar@{-}[dl]\\ PO_N^+\ar@{-}[u]\ar@{-}[ur]\ar@{-}[r]&PH_N^+\ar@{-}[l]\ar@{-}[ul]\ar@{-}[u]} [[/math]]

The question is that of axiomatizing these quadruplets.

To be more precise, in analogy with what happens in the affine case, the problem is that of establishing correspondences as follows:

[[math]] \xymatrix@R=50pt@C=50pt{ P\ar[r]\ar[d]\ar[dr]&PT\ar[l]\ar[d]\ar[dl]\\ PU\ar[u]\ar[ur]\ar[r]&PK\ar[l]\ar[ul]\ar[u] } [[/math]]

Modulo this problem, which is for the moment open, things are potentially quite nice, because we seem to have only 3 geometries, namely real, complex and free. Generally speaking, we are led in this way into several questions:

(1) We first need functoriality results for the operations [math] \lt \,, \gt [/math] and [math]\cap[/math], in relation with taking the projective version, and taking affine lifts, as to deduce most of our 7 axioms, in their obvious projective formulation, from the affine ones.

(2) Then, we need quantum isometry results in the projective setting, for the projective spaces themselves, and for the projective tori, either established ad-hoc, or by using the affine results. For the projective spaces, this was done above.

(3) We need as well some further functoriality results, in order to axiomatize the intermediate objects that we are dealing, the problem here being whether we want to use projective objects, or projective versions, perhaps saturated, of affine objects.

(4) Modulo this, things are quite clear, with the final result being the fact that we have only 3 projective geometries. Technically, the proof should be using the fact that, in the easy setting, [math]PO_N\subset PU_N\subset PO_N^+[/math] are the only possible unitary groups.

Let us also mention that, in the noncommutative setting, there are several ways of defining the projective versions, with the one used here being the “simplest”. As explained in , , it is possible to construct a left projective version, a right projective version, and a mixed projective version, with all these operations being interesting. Thus, the results and problems presented above are just the “tip of the iceberg”, with the general projective space and version problematics being much wider then this.

As another remark, our results tend to suggest that the free projective geometry is “scalarless”. However, things here are quite complicated, because, while there has been some interesting preliminary work on this subject by Bichon and others, it is still not presently known what easiness should mean, over an arbitrary field [math]F[/math].

Finally, and above everything, the free projective geometry remains to be developed. A first piece of homework here is that of developing a theory of free Grassmannians, free flag manifolds, and free Stiefel manifolds, based on the affine theory of the spaces of quantum partial isometries, developed in chapter 6. To be more precise, the definition of the free Grassmannians is straightforward, as follows, and the definition of the free flag manifolds and free Stiefel manifolds is most likely something very similar:

[[math]] C(Gr_{LN}^+)=C^*\left((p_{ij})_{i,j=1,\ldots,N}\Big|p=p^*=p^2,Tr(p)=L\right) [[/math]]

Things do not look difficult here, with most of the arguments from the affine case carrying over in the projective setting, and with solid affine results to rely upon being available from chapter 6. But work to be done for sure, which has not been done yet.

General references

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