2b. Free rotations

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Let us get back now to our original objective, namely constructing pairs of quantum unitary and reflection groups [math](O_N^+,H_N^+)[/math] and [math](U_N^+,K_N^+)[/math], as to complete the pairs [math](S^{N-1}_{\mathbb R,+},T_N^+)[/math] and [math](S^{N-1}_{\mathbb C,+},\mathbb T_N^+)[/math] that we already have. Following Wang , we have:

Theorem

The following constructions produce compact quantum groups,

[[math]] \begin{eqnarray*} C(O_N^+)&=&C^*\left((u_{ij})_{i,j=1,\ldots,N}\Big|u=\bar{u},u^t=u^{-1}\right)\\ C(U_N^+)&=&C^*\left((u_{ij})_{i,j=1,\ldots,N}\Big|u^*=u^{-1},u^t=\bar{u}^{-1}\right) \end{eqnarray*} [[/math]]

which appear respectively as liberations of the groups [math]O_N[/math] and [math]U_N[/math].

Show Proof

This first assertion follows from the elementary fact that if a matrix [math]u=(u_{ij})[/math] is orthogonal or biunitary, then so must be the following matrices:

[[math]] u^\Delta_{ij}=\sum_ku_{ik}\otimes u_{kj} [[/math]]

[[math]] u^\varepsilon_{ij}=\delta_{ij} [[/math]]

[[math]] u^S_{ij}=u_{ji}^* [[/math]]

Indeed, the biunitarity of [math]u^\Delta[/math] can be checked by a direct computation. Regarding now the matrix [math]u^\varepsilon=1_N[/math], this is clearly biunitary. Also, regarding the matrix [math]u^S[/math], there is nothing to prove here either, because its unitarity its clear too. And finally, observe that if [math]u[/math] has self-adjoint entries, then so do the above matrices [math]u^\Delta,u^\varepsilon,u^S[/math].

Thus our claim is proved, and we can define morphisms [math]\Delta,\varepsilon,S[/math] as in Definition 2.1, by using the universal properties of [math]C(O_N^+)[/math], [math]C(U_N^+)[/math]. As for the second assertion, this follows exactly as for the free spheres, by adapting the sphere proof from chapter 1.

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The basic properties of [math]O_N^+,U_N^+[/math] can be summarized as follows:

Theorem

The quantum groups [math]O_N^+,U_N^+[/math] have the following properties:

• The closed subgroups [math]G\subset U_N^+[/math] are exactly the [math]N\times N[/math] compact quantum groups. As for the closed subgroups [math]G\subset O_N^+[/math], these are those satisfying [math]u=\bar{u}[/math].
• We have liberation embeddings [math]O_N\subset O_N^+[/math] and [math]U_N\subset U_N^+[/math], obtained by dividing the algebras [math]C(O_N^+),C(U_N^+)[/math] by their respective commutator ideals.
• We have as well embeddings [math]\widehat{L}_N\subset O_N^+[/math] and [math]\widehat{F}_N\subset U_N^+[/math], where [math]L_N[/math] is the free product of [math]N[/math] copies of [math]\mathbb Z_2[/math], and where [math]F_N[/math] is the free group on [math]N[/math] generators.

Show Proof

All these assertions are elementary, as follows:

(1) This is clear from definitions, with the remark that, in the context of Definition 2.1, the formula [math]S(u_{ij})=u_{ji}^*[/math] shows that the matrix [math]\bar{u}[/math] must be unitary too.

(2) This follows from the Gelfand theorem. To be more precise, this shows that we have presentation results for [math]C(O_N),C(U_N)[/math], similar to those in Theorem 2.12, but with the commutativity between the standard coordinates and their adjoints added:

[[math]] \begin{eqnarray*} C(O_N)&=&C^*_{comm}\left((u_{ij})_{i,j=1,\ldots,N}\Big|u=\bar{u},u^t=u^{-1}\right)\\ C(U_N)&=&C^*_{comm}\left((u_{ij})_{i,j=1,\ldots,N}\Big|u^*=u^{-1},u^t=\bar{u}^{-1}\right) \end{eqnarray*} [[/math]]

Thus, we are led to the conclusion in the statement.

(3) This follows indeed from (1) and from Theorem 2.2, with the remark that with [math]u=diag(g_1,\ldots,g_N)[/math], the condition [math]u=\bar{u}[/math] is equivalent to [math]g_i^2=1[/math], for any [math]i[/math].

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The last assertion in Theorem 2.13 suggests the following construction:

Proposition

Given a closed subgroup [math]G\subset U_N^+[/math], consider its “diagonal torus”, which is the closed subgroup [math]T\subset G[/math] constructed as follows:

[[math]] C(T)=C(G)\Big/\left \lt u_{ij}=0\Big|\forall i\neq j\right \gt [[/math]]

This torus is then a group dual, [math]T=\widehat{\Lambda}[/math], where [math]\Lambda= \lt g_1,\ldots,g_N \gt [/math] is the discrete group generated by the elements [math]g_i=u_{ii}[/math], which are unitaries inside [math]C(T)[/math].

Show Proof

Since [math]u[/math] is unitary, its diagonal entries [math]g_i=u_{ii}[/math] are unitaries inside [math]C(T)[/math]. Moreover, from [math]\Delta(u_{ij})=\sum_ku_{ik}\otimes u_{kj}[/math] we obtain, when passing inside the quotient:

[[math]] \Delta(g_i)=g_i\otimes g_i [[/math]]

It follows that we have [math]C(T)=C^*(\Lambda)[/math], modulo identifying as usual the [math]C^*[/math]-completions of the various group algebras, and so that we have [math]T=\widehat{\Lambda}[/math], as claimed.

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With this notion in hand, Theorem 2.13 (3) reformulates as follows:

Theorem

The diagonal tori of the basic unitary groups are the basic tori:

[[math]] \xymatrix@R=16.5mm@C=18mm{ O_N^+\ar[r]&U_N^+\\ O_N\ar[r]\ar[u]&U_N\ar[u]} \qquad \item[a]ymatrix@R=8mm@C=15mm{\\ \to} \qquad \item[a]ymatrix@R=16.5mm@C=18mm{ T_N^+\ar[r]&\mathbb T_N^+\\ T_N\ar[r]\ar[u]&\mathbb T_N\ar[u]} [[/math]]

In particular, the basic unitary groups are all distinct.

Show Proof

This is something clear and well-known in the classical case, and in the free case, this is a reformulation of Theorem 2.13 (3), which tells us that the diagonal tori of [math]O_N^+,U_N^+[/math], in the sense of Proposition 2.14, are the group duals [math]\widehat{L}_N,\widehat{F}_N[/math].

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There is an obvious relation here with the considerations from chapter 1, that we will analyse later on. As a second result now regarding our free quantum groups, relating them this time to the free spheres constructed in chapter 1, we have:

Proposition

We have embeddings of algebraic manifolds as follows, obtained in double indices by rescaling the coordinates, [math]x_{ij}=u_{ij}/\sqrt{N}[/math]:

[[math]] \xymatrix@R=16.5mm@C=18mm{ O_N^+\ar[r]&U_N^+\\ O_N\ar[r]\ar[u]&U_N\ar[u]} \qquad \item[a]ymatrix@R=8mm@C=5mm{\\ \to} \qquad \item[a]ymatrix@R=15mm@C=14mm{ S^{N^2-1}_{\mathbb R,+}\ar[r]&S^{N^2-1}_{\mathbb C,+}\\ S^{N^2-1}_\mathbb R\ar[r]\ar[u]&S^{N^2-1}_\mathbb C\ar[u] } [[/math]]

Moreover, the quantum groups appear from the quantum spheres via

[[math]] G=S\cap U_N^+ [[/math]]

with the intersection being computed inside the free sphere [math]S^{N^2-1}_{\mathbb C,+}[/math].

Show Proof

As explained in Theorem 2.11, the biunitarity of the matrix [math]u=(u_{ij})[/math] gives an embedding of algebraic manifolds, as follows:

[[math]] U_N^+\subset S^{N^2-1}_{\mathbb C,+} [[/math]]

Now since the relations defining [math]O_N,O_N^+,U_N\subset U_N^+[/math] are the same as those defining [math]S^{N^2-1}_\mathbb R,S^{N^2-1}_{\mathbb R,+},S^{N^2-1}_\mathbb C\subset S^{N^2-1}_{\mathbb C,+}[/math], this gives the result.

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General references

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