6d. Further results

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Let us discuss now the relation with [math]O_N^+[/math]. As explained earlier in this chapter, in the classical case the passage [math]O_N\to U_N[/math] is something not trivial, requiring a passage via the associated Lie algebras. In the free case the situation is very simple, as follows:

Theorem

We have an identification as follows,

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

modulo the usual equivalence relation for compact quantum groups.

Show Proof

We recall from chapter 2 that the free complexification operation [math]G\to\widetilde{G}[/math] is obtained by multiplying the coefficients of the fundamental representation by a unitary free from them. 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) The original argument, from , is something quick and advanced, based on the standard free probability fact that 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, exactly as for [math]U_N^+[/math], and 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 this proof, not using any prior free probability knowledge, is by using fusion rules. Indeed, as explained in chapter 2 above, the representations of the dual free products, and in particular of the free complexifications, can be explicitely computed. Thus the fusion rules for [math]\widetilde{O_N^+}[/math] appear as a “free complexification” of the Clebsch-Gordan rules for [math]O_N^+[/math], and in practice this leads to the same fusion rules as for [math]U_N^+[/math]. As before, by Peter-Weyl we obtain from this that the inclusion [math]\widetilde{O_N^+}\subset U_N^+[/math] must be an isomorphism, modulo the usual equivalence relation for the compact quantum groups.

(3) A third proof of the result, based on the same idea, and which is perhaps the simplest, makes use of the easiness property of [math]O_N^+,U_N^+[/math] only. Indeed, let us denote by [math]v,zv,u[/math] the fundamental representations of the following 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 chapter 4 above, the result there stating 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 a comment here, the proof (3) above, when properly worked out, provides as well an alternative proof for Theorem 6.20. Indeed, once we know that we have [math]U_N^+=\widetilde{O_N^+}[/math], it follows that the fusion rules for [math]U_N^+[/math] appear as a “free complexification” of the Clebsch-Gordan rules for [math]O_N^+[/math], and in practice this leads to the formulae in Theorem 6.20.

However, this is nowhere done in the literature, and if you prefer this kind of proof, which is purely algebraic, you will have to work it out by yourself. The problem is that, with this proof, you still have to show afterwards that [math]\chi[/math] is circular, and this is best done starting from [math]U_N^+=\widetilde{O_N^+}[/math], and using the polar decomposition of circular variables, which is a free probability result due to Voiculescu , which is not exactly trivial.

Let us summarize this discussion by recording the following fact: \begin{fact} It is possible to establish the main results regarding [math]U_N^+[/math], namely

• Free compexification, [math]U_N^+=\widetilde{O_N^+}[/math]
• Fusion rules, [math]r_k\otimes r_l=\sum_{k=xy,l=\bar{y}z}r_{xz}[/math]
• Character law, [math]\chi\sim\Gamma_1[/math]

by using diagrams for [math](1)[/math], and then proving [math](1)\implies(2),(3)[/math]. \end{fact} Which leads us into the question on why was not written in this way, because that was a research paper, where the use of anything from was allowed anyway. Well, the story here is that was my PhD thesis, and my advisor Georges Skandalis, as one of the main architects, with his colleague Saad Baaj, of the theory of locally compact quantum groups with [math]S^2\neq id[/math], was insisting for me to do the work for the [math]S^2\neq id[/math] analogues of [math]U_N^+[/math] too, and for certain technical reasons, this cannot be done as in Fact 6.22.

In short, and as an advice now if you are a PhD student, just shut up and do what your advisor is saying, as a perfect mercenary. Discipline first, and learn to kill anything upon request, that's always a good skill to have. And plenty of time later to fully express yourself, during a long career. As I am actually doing myself now, when writing this book, with [math]S^2=id[/math] as an axiom, contrary to what Georges and Saad have taught me.

Back to work now, 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 6.21, 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.

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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.

More generally, once again by following , we have similar results obtained by replacing [math]O_N^+[/math] with the more general super-orthogonal quantum groups [math]O_F^+[/math] from the previous chapter, which include as well the free symplectic groups [math]Sp_N^+[/math]. Let us start with:

Theorem

We have an identification as follows,

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

valid for any super-orthogonal quantum group [math]O_F^+[/math].

Show Proof

This is a straightforward extension of Theorem 6.21 above, with any of the proofs there extending to the case of the quantum groups [math]O_F^+[/math]. See .

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We have as well a projective version of the above result, as follows:

Theorem

We have an identification as follows,

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

valid for any super-orthogonal quantum group [math]O_F^+[/math].

Show Proof

This is a straightforward extension of Theorem 6.23, with any of the proofs there extending to the case of the quantum groups [math]O_F^+[/math]. Alternatively, the result follows from Theorem 6.24, by taking the projective versions of the quantum groups there.

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The free symplectic result at [math]N=2[/math] is particularly interesting, because here we have [math]Sp_2^+=SU_2[/math], and so we obtain that [math]U_2^+[/math] is the free complexification of [math]SU_2[/math]:

Theorem

We have an identification as follows,

[[math]] U_2^+=\widetilde{SU_2} [[/math]]

modulo the usual equivalence relation for compact quantum groups.

Show Proof

As explained above, this follows from Theorem 6.24, and from [math]Sp_2^+=SU_2[/math], via the material explained in chapter 5 above. See , .

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Finally, we have a projective version of the above result, as follows:

Theorem

We have an identification as follows, and this even without using the standard equivalence relation for the compact quantum groups:

[[math]] PU_2^+=SO_3 [[/math]]

A similar result holds for the “left” projective version of [math]U_2^+[/math], constructed by using the corepresentation [math]\bar{u}\otimes u[/math] instead of [math]u\otimes\bar{u}[/math].

Show Proof

We have several assertions here, the idea being as follows:

(1) By using Theorem 6.26 we obtain, modulo the equivalence relation:

[[math]] PU_2^+=P\widetilde{SU_2}=PSU_2=SO_3 [[/math]]

(2) Now since [math]SO_3[/math] is coamenable, the above formula must hold in fact in a plain way, meaning without using the equivalence relation. This can be checked as well directly, by verifying that the coefficients of [math]u\otimes\bar{u}[/math] commute indeed.

(3) Finally, the last assertion can be either deduced from the first one, or proved directly, by using “left” free complexification operations, in all the above.

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We refer to for further applications of the above [math]N=2[/math] results, for instance with structure results regarding the von Neumann algebra [math]L^\infty(U_2^+)[/math]. We will be back to [math]U_N^+[/math] in chapter 8 below, with a number of more advanced probabilistic results about it.

General references

Banica, Teo (2024). "Introduction to quantum groups". arXiv:1909.08152 [math.CO].