guide:0ae8b28bc6

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

This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.

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.

■

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].

■

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.

■

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].

■

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.

■

General references

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