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This chapter is an introduction to projective geometry, in our sense. The point is that things become considerably simpler in the projective setting. Consider indeed the diagram of 9 main geometries, that we found above:

[[math]] \xymatrix@R=40pt@C=42pt{ \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]]

As explained in chapters 9-10, when looking at the projective versions of these geometries, the diagram drastically simplifies. To be more precise, the diagram of projective versions of the corresponding spheres are as follows, consisting of 3 objects only:

[[math]] \xymatrix@R=34pt@C=33pt{ P^{N-1}_+\ar[r]&P^{N-1}_+\ar[r]&P^{N-1}_+\\ P^{N-1}_\mathbb C\ar[r]\ar[u]&P^{N-1}_\mathbb C\ar[r]\ar[u]&P^{N-1}_\mathbb C\ar[u]\\ P^{N-1}_\mathbb R\ar[r]\ar[u]&P^{N-1}_\mathbb R\ar[r]\ar[u]&P^{N-1}_\mathbb C\ar[u]} [[/math]]

Thus, we are led to the conclusion that, under certain combinatorial axioms, there should be only 3 projective geometries, namely the real, complex and free ones:

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

We will discuss this in what follows, with analogues and improvements of the affine results. Also, we would like to study the corresponding quadruplets [math](P,PT,PU,PK)[/math], and to axiomatize the projective geometries, with 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]]

Summarizing, there is a lot of work to be done, on one hand in reformulating and improving the results from the affine case, and on the other hand, in starting to develop the projective theory independently from the affine theory.

As a first topic that we would like to discuss, which historically speaking, was at the beginning of everything, we have the following remarkable isomorphism, that we already used in the above, and that we would like to discuss now in detail:

[[math]] PO_N^+=PU_N^+ [[/math]]

In order to get started, let us first discuss the classical case, and more specifically the precise relation between the orthogonal group [math]O_N[/math], and the unitary group [math]U_N[/math]. Contrary to the passage [math]\mathbb R^N\to\mathbb C^N[/math], or to the passage [math]S^{N-1}_\mathbb R\to S^{N-1}_\mathbb C[/math], which are both elementary, the passage [math]O_N\to U_N[/math] cannot be understood directly. In order to understand this passage we must pass through the corresponding Lie algebras, a follows:

Theorem

The passage [math]O_N\to U_N[/math] appears via Lie algebra complexification,

[[math]] O_N\to\mathfrak o_N\to\mathfrak u_n\to U_N [[/math]]

with the Lie algebra [math]\mathfrak u_N[/math] being a complexification of the Lie algebra [math]\mathfrak o_N[/math].

Show Proof

This is something rather philosophical, and advanced as well, that we will not really need here, the idea being as follows:

(1) The unitary and orthogonal groups [math]U_N,O_N[/math] are both Lie groups, in the sense that they are smooth manifolds. The corresponding Lie algebras [math]\mathfrak u_N,\mathfrak o_N[/math], which are by definition the respective tangent spaces at 1, can be computed by differentiating the equations defining [math]U_N,O_N[/math], with the conclusion being as follows:

[[math]] \mathfrak u_N=\left\{ A\in M_N(\mathbb C)\Big|A^*=-A\right\} [[/math]]

[[math]] \mathfrak o_N=\left\{ B\in M_N(\mathbb R)\Big|B^t=-B\right\} [[/math]]

(2) This was for the correspondences [math]U_N\to\mathfrak u_N[/math] and [math]O_N\to\mathfrak o_N[/math]. In the other sense, the correspondences [math]\mathfrak u_N\to U_N[/math] and [math]\mathfrak o_N\to O_N[/math] appear by exponentiation, the result here stating that, around 1, the unitary matrices can be written as [math]U=e^A[/math], with [math]A\in\mathfrak u_N[/math], and the orthogonal matrices can be written as [math]U=e^B[/math], with [math]B\in\mathfrak o_N[/math].

(3) In view of all this, in order to understand the passage [math]O_N\to U_N[/math] it is enough to understand the passage [math]\mathfrak o_N\to\mathfrak u_N[/math]. But, in view of the above formulae for [math]\mathfrak o_N,\mathfrak u_N[/math], this is basically an elementary linear algebra problem. Indeed, let us pick an arbitrary matrix [math]A\in M_N(\mathbb C)[/math], and write it as follows, with [math]B,C\in M_N(\mathbb R)[/math]:

[[math]] A=B+iC [[/math]]

In terms of [math]B,C[/math], the equation [math]A^*=-A[/math] defining the Lie algebra [math]\mathfrak u_N[/math] reads:

[[math]] B^t=-B\quad,\quad C^t=C [[/math]]

(4) As a first observation, we must have [math]B\in\mathfrak o_N[/math]. Regarding now [math]C[/math], let us decompose this matrix as follows, with [math]D[/math] being its diagonal, and [math]C'[/math] being the reminder:

[[math]] C=D+C' [[/math]]

The matrix [math]C'[/math] being symmetric with 0 on the diagonal, by swithcing all the signs below the main diagonal we obtain a certain matrix [math]C'_-\in\mathfrak o_N[/math]. Thus, we have decomposed [math]A\in\mathfrak u_N[/math] as follows, with [math]B,C'\in\mathfrak o_N[/math], and with [math]D\in M_N(\mathbb R)[/math] being diagonal:

[[math]] A=B+iD+iC'_- [[/math]]

(5) As a conclusion now, we have shown that we have a direct sum decomposition of real linear spaces as follows, with [math]\Delta\subset M_N(\mathbb R)[/math] being the diagonal matrices:

[[math]] \mathfrak u_N\simeq\mathfrak o_N\oplus\Delta\oplus\mathfrak o_N [[/math]]

Thus, we can stop our study here, and say that we have reached the conclusion in the statement, namely that [math]\mathfrak u_N[/math] appears as a “complexification” of [math]\mathfrak o_N[/math].

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As before with many other things, that we will not really need in what follows, this was just an introduction to the subject. More can be found in any Lie group book. In the free case now, the situation is much simpler, and we have:

Theorem

The passage [math]O_N^+\to U_N^+[/math] appears via free complexification,

[[math]] U_N^+=\widetilde{O_N^+} [[/math]]

where the free complexification of a pair [math](G,u)[/math] is the pair [math](\widetilde{G},\widetilde{u})[/math] with

[[math]] C(\widetilde{G})= \lt zu_{ij} \gt \subset C(\mathbb T)*C(G)\quad,\quad \widetilde{u}=zu [[/math]]

where [math]z\in C(\mathbb T)[/math] is the standard generator, given by [math]x\to x[/math] for any [math]x\in\mathbb T[/math].

Show Proof

We have embeddings as follows, with the first one coming by using the counit, and with the second one coming from the universality property of [math]U_N^+[/math]:

[[math]] O_N^+ \subset\widetilde{O_N^+} \subset U_N^+ [[/math]]

We must prove that the embedding on the right is an isomorphism, and there are several ways of doing this, all instructive, as follows:

(1) If we denote by [math]v,u[/math] the fundamental corepresentations of [math]O_N^+,U_N^+[/math], we have:

[[math]] Fix(v^{\otimes k})=span\left(\xi_\pi\Big|\pi\in NC_2(k)\right) [[/math]]

[[math]] Fix(u^{\otimes k})=span\left(\xi_\pi\Big|\pi\in\mathcal{NC}_2(k)\right) [[/math]]

Moreover, the above vectors [math]\xi_\pi[/math] are known to be linearly independent at [math]N\geq2[/math], and so the above results provide us with bases, and we obtain:

[[math]] \dim(Fix(v^{\otimes k}))=|NC_2(k)|\quad,\quad \dim(Fix(u^{\otimes k}))=|\mathcal{NC}_2(k)| [[/math]]

Now since integrating the character of a corepresentation amounts in counting the fixed points, the above two formulae can be rewritten as follows:

[[math]] \int_{O_N^+}\chi_v^k=|NC_2(k)|\quad,\quad \int_{U_N^+}\chi_u^k=|\mathcal{NC}_2(k)| [[/math]]

But this shows, via standard free probability theory, that [math]\chi_v[/math] must follow the Winger semicircle law [math]\gamma_1[/math], and that [math]\chi_u[/math] must follow the Voiculescu circular law [math]\Gamma_1[/math]:

[[math]] \chi_v\sim\gamma_1\quad,\quad \chi_u\sim\Gamma_1 [[/math]]

On the other hand, by , when freely multiplying a semicircular variable by a Haar unitary we obtain a circular variable. Thus, the main character of [math]\widetilde{O_N^+}[/math] is circular:

[[math]] \chi_{zv}\sim\Gamma_1 [[/math]]

Now by forgetting about circular variables and free probability, the conclusion is that the inclusion [math]\widetilde{O_N^+}\subset U_N^+[/math] preserves the law of the main character:

[[math]] law(\chi_{zv})=law(u) [[/math]]

Thus by Peter-Weyl we obtain that the inclusion [math]\widetilde{O_N^+}\subset U_N^+[/math] must be an isomorphism, modulo the usual equivalence relation for quantum groups.

(2) A version of the above proof, not using any prior free probability knowledge, makes use of the easiness property of [math]O_N^+,U_N^+[/math] only, namely:

[[math]] Hom(v^{\otimes k},v^{\otimes l})=span\left(\xi_\pi\Big|\pi\in NC_2(k,l)\right) [[/math]]

[[math]] Hom(u^{\otimes k},u^{\otimes l})=span\left(\xi_\pi\Big|\pi\in\mathcal{NC}_2(k,l)\right) [[/math]]

Indeed, let us look at the following inclusions of quantum groups:

[[math]] O_N^+\subset\widetilde{O_N^+}\subset U_N^+ [[/math]]

At the level of the associated Hom spaces we obtain reverse inclusions, as follows:

[[math]] Hom(v^{\otimes k},v^{\otimes l}) \supset Hom((zv)^{\otimes k},(zv)^{\otimes l}) \supset Hom(u^{\otimes k},u^{\otimes l}) [[/math]]

The spaces on the left and on the right are known from easiness, the result being that these spaces are as follows:

[[math]] span\left(T_\pi\Big|\pi\in NC_2(k,l)\right)\supset span\left(T_\pi\Big|\pi\in\mathcal{NC}_2(k,l)\right) [[/math]]

Regarding the spaces in the middle, these are obtained from those on the left by “coloring”, so we obtain the same spaces as those on the right. Thus, by Tannakian duality, our embedding [math]\widetilde{O_N^+}\subset U_N^+[/math] is an isomorphism, modulo the usual equivalence relation.

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As an interesting consequence of the above result, we have:

Theorem

We have an identification as follows,

[[math]] PO_N^+=PU_N^+ [[/math]]

modulo the usual equivalence relation for compact quantum groups.

Show Proof

As before, we have several proofs for this result, as follows:

(1) This follows from Theorem 15.2, because we have:

[[math]] PU_N^+=P\widetilde{O_N^+}=PO_N^+ [[/math]]

(2) We can deduce this as well directly. With notations as before, we have:

[[math]] Hom\left((v\otimes v)^k,(v\otimes v)^l\right)=span\left(T_\pi\Big|\pi\in NC_2((\circ\bullet)^k,(\circ\bullet)^l)\right) [[/math]]

[[math]] Hom\left((u\otimes\bar{u})^k,(u\otimes\bar{u})^l\right)=span\left(T_\pi\Big|\pi\in \mathcal{NC}_2((\circ\bullet)^k,(\circ\bullet)^l)\right) [[/math]]

The sets on the right being equal, we conclude that the inclusion [math]PO_N^+\subset PU_N^+[/math] preserves the corresponding Tannakian categories, and so must be an isomorphism.

■

As a conclusion, the passage [math]O_N^+\to U_N^+[/math] is something much simpler than the passage [math]O_N\to U_N[/math], with this ultimately coming from the fact that the combinatorics of [math]O_N^+,U_N^+[/math] is something much simpler than the combinatorics of [math]O_N,U_N[/math]. In addition, all this leads as well to the interesting conclusion that the free projective geometry does not fall into real and complex, but is rather unique and “scalarless”. We will be back to this.

Let us discuss now the projective spaces. We begin with a short summary of the various projective geometry results that we have so far. We will give full details here, with the aim of making the present chapter as independent as possible from the previous chapters, as a beginning of something new. Our starting point is the following functional analytic description of the real and complex projective spaces [math]P^{N-1}_\mathbb R,P^{N-1}_\mathbb C[/math]:

Proposition

We have presentation results as follows,

[[math]] \begin{eqnarray*} C(P^{N-1}_\mathbb R)&=&C^*_{comm}\left((p_{ij})_{i,j=1,\ldots,N}\Big|p=\bar{p}=p^t=p^2,Tr(p)=1\right)\\ C(P^{N-1}_\mathbb C)&=&C^*_{comm}\left((p_{ij})_{i,j=1,\ldots,N}\Big|p=p^*=p^2,Tr(p)=1\right) \end{eqnarray*} [[/math]]

for the algebras of continuous functions on the real and complex projective spaces.

Show Proof

We use the fact that the projective spaces [math]P^{N-1}_\mathbb R,P^{N-1}_\mathbb C[/math] can be respectively identified with the spaces of rank one projections in [math]M_N(\mathbb R),M_N(\mathbb C)[/math]. With this picture in mind, it is clear that we have arrows [math]\leftarrow[/math]. In order to construct now arrows [math]\to[/math], consider the universal algebras on the right, [math]A_R,A_C[/math]. These algebras being both commutative, by the Gelfand theorem we can write, with [math]X_R,X_C[/math] being certain compact spaces:

[[math]] A_R=C(X_R)\quad,\quad A_C=C(X_C) [[/math]]

Now by using the coordinate functions [math]p_{ij}[/math], we conclude that [math]X_R,X_C[/math] are certain spaces of rank one projections in [math]M_N(\mathbb R),M_N(\mathbb C)[/math]. In other words, we have embeddings:

[[math]] X_R\subset P^{N-1}_\mathbb R\quad,\quad X_C\subset P^{N-1}_\mathbb C [[/math]]

By transposing we obtain arrows [math]\to[/math], as desired.

■

The above result suggests the following definition:

Definition

Associated to any [math]N\in\mathbb N[/math] is the following universal algebra,

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

whose abstract spectrum is called “free projective space”.

Observe that, according to our presentation results for the real and complex projective spaces [math]P^{N-1}_\mathbb R[/math] and [math]P^{N-1}_\mathbb C[/math], we have embeddings of compact quantum spaces, as follows:

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

Let us first discuss the relation with the spheres. Given a closed subset [math]X\subset S^{N-1}_{\mathbb R,+}[/math], its projective version is by definition the quotient space [math]X\to PX[/math] determined by the fact that [math]C(PX)\subset C(X)[/math] is the subalgebra generated by the following variables:

[[math]] p_{ij}=x_ix_j [[/math]]

In order to discuss the relation with the spheres, it is convenient to neglect the material regarding the complex and hybrid cases, the projective versions of such spheres bringing nothing new. Thus, we are left with the 3 real spheres, and we have:

Proposition

The projective versions of the [math]3[/math] real spheres are as follows,

[[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]]

modulo the standard equivalence relation for the quantum algebraic manifolds.

Show Proof

The assertion at left is true by definition. For the assertion at right, we have to prove that the variables [math]p_{ij}=z_iz_j[/math] over the free sphere [math]S^{N-1}_{\mathbb R,+}[/math] satisfy the defining relations for [math]C(P^{N-1}_+)[/math], from Definition 15.5, namely:

[[math]] p=p^*=p^2\quad,\quad Tr(p)=1 [[/math]]

We first have the following computation:

[[math]] (p^*)_{ij} =p_{ji}^* =(z_jz_i)^* =z_iz_j =p_{ij} [[/math]]

We have as well the following computation:

[[math]] (p^2)_{ij} =\sum_kp_{ik}p_{kj} =\sum_kz_iz_k^2z_j =z_iz_j\\ =p_{ij} [[/math]]

Finally, we have as well the following computation:

[[math]] Tr(p) =\sum_kp_{kk} =\sum_kz_k^2 =1 [[/math]]

Regarding now [math]PS^{N-1}_{\mathbb R,*}=P^{N-1}_\mathbb C[/math], the inclusion “[math]\subset[/math]” follows from [math]abcd=cbad=cbda[/math]. In the other sense now, the point is that we have a matrix model, as follows:

[[math]] \pi:C(S^{N-1}_{\mathbb R,*})\to M_2(C(S^{N-1}_\mathbb C))\quad,\quad x_i\to\begin{pmatrix}0&z_i\\ \bar{z}_i&0\end{pmatrix} [[/math]]

But this gives the missing inclusion “[math]\supset[/math]”, and we are done. See .

■

In addition to the above result, let us mention that, as already discussed above, passing to the complex case brings nothing new. This is because the projective version of the free complex sphere is equal to the free projective space constructed above:

[[math]] PS^{N-1}_{\mathbb C,+}=P^{N-1}_+ [[/math]]

And the same goes for the “hybrid” spheres. For details on all this, we refer to chapters 9-10. In what regards now the tori, we have here the following result:

Proposition

The projective versions of the [math]3[/math] real tori are as follows,

[[math]] \xymatrix@R=15mm@C=15mm{ T_N\ar[r]\ar[d]&T_N^*\ar[r]\ar[d]&T_N^+\ar[d]\\ PT_N\ar[r]&P\mathbb T_N\ar[r]&PT_N^+} [[/math]]

modulo the standard equivalence relation for the quantum algebraic manifolds.

Show Proof

This follows indeed by using the same arguments as for the spheres.

■

In what regards the orthogonal groups, we have here the following result:

Proposition

The projective versions of the [math]3[/math] orthogonal groups are

[[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]]

modulo the standard equivalence relation for the compact quantum groups.

Show Proof

This follows by using the same arguments as for spheres, or tori.

■

Finally, in what regards the reflection groups, we have here the following result:

Proposition

The projective versions of the [math]3[/math] reflection groups are

[[math]] \xymatrix@R=15mm@C=15mm{ H_N\ar[r]\ar[d]&H_N^*\ar[r]\ar[d]&H_N^+\ar[d]\\ PH_N\ar[r]&PK_N\ar[r]&PH_N^+} [[/math]]

modulo the standard equivalence relation for the compact quantum groups.

Show Proof

This follows indeed by using the same arguments as before.

■

Let us mention that, as it was the case for the spheres, passing to the complex case brings nothing new. This is indeed because we have isomorphisms as follows, which can be established by using easiness, as explained in the beginning of the present chapter for the isomorphism in the middle, and with the proof of the first and of the last isomorphism being quite similar, based respectively on elementary group theory, and on easiness:

[[math]] P\mathbb T_N^+=PT_N^+\quad,\quad PU_N^+=PO_N^+\quad,\quad PK_N^+=PH_N^+ [[/math]]

Thus, passing to the complex case would bring indeed nothing new, and in what follows we will stay with the real formalism. This is of course something quite subtle, which happens only in the free case, and has no classical counterpart. For further details and comments here, we refer to the discussion in the beginning of this chapter.

Getting back now to our general program, we are done with the construction work, for the various projective geometry basic objects. As a conclusion to what we did, with our above constructions, in the projective geometry setting, we have 3 projective quadruplets, whose construction and main properties can be summarized as follows:

Theorem

We have projective quadruplets [math](P,PT,PU,PK)[/math] as follows,

• 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]]
• A classical complex quadruplet, 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]]
• A free quadruplet, 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]]

which appear as projective versions of the main [math]3[/math] real quadruplets.

Show Proof

This follows indeed from the results that already have. To be more precise, the details, that we will we need in what comes next, are as follows:

(1) Consider the classical affine real quadruplet, which is as follows:

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

The projective version of this quadruplet is then the quadruplet in (1).

(2) Consider the half-classical affine real quadruplet, which is as follows:

[[math]] \xymatrix@R=50pt@C=50pt{ S^{N-1}_{\mathbb R,*}\ar@{-}[r]\ar@{-}[d]\ar@{-}[dr]&T_N^*\ar@{-}[l]\ar@{-}[d]\ar@{-}[dl]\\ O_N^*\ar@{-}[u]\ar@{-}[ur]\ar@{-}[r]&H_N^*\ar@{-}[l]\ar@{-}[ul]\ar@{-}[u]} [[/math]]

The projective version of this quadruplet is then the quadruplet in (2).

(3) Consider the free affine real quadruplet, which is as follows:

[[math]] \xymatrix@R=50pt@C=50pt{ S^{N-1}_{\mathbb R,+}\ar@{-}[r]\ar@{-}[d]\ar@{-}[dr]&T_N^+\ar@{-}[l]\ar@{-}[d]\ar@{-}[dl]\\ O_N^+\ar@{-}[u]\ar@{-}[ur]\ar@{-}[r]&H_N^+\ar@{-}[l]\ar@{-}[ul]\ar@{-}[u]} [[/math]]

The projective version of this quadruplet is then the quadruplet in (3).

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General references

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