2d. Free constructions

[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.

At the level of the really “new” examples now, we have basic liberation constructions, going back to the pioneering work of Wang , , and to the subsequent papers , , as well as several more recent constructions. We first have, following Wang :

Theorem

The following universal algebras are Woronowicz algebras,

[[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]]

so the underlying compact quantum spaces [math]O_N^+,U_N^+[/math] are compact quantum groups.

Show Proof

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

[[math]] u^\Delta_{ij}=\sum_ku_{ik}\otimes u_{kj}\quad,\quad u^\varepsilon_{ij}=\delta_{ij}\quad,\quad u^S_{ij}=u_{ji}^* [[/math]]

Consider indeed the matrix [math]U=u^\Delta[/math]. We have then:

[[math]] (UU^*)_{ij} =\sum_{klm}u_{il}u_{jm}^*\otimes u_{lk}u_{mk}^* =\sum_{lm}u_{il}u_{jm}^*\otimes\delta_{lm} =\delta_{ij} [[/math]]

In the other sense the computation is similar, as follows:

[[math]] (U^*U)_{ij} =\sum_{klm}u_{kl}^*u_{km}\otimes u_{li}^*u_{mj} =\sum_{lm}\delta_{lm}\otimes u_{li}^*u_{mj} =\delta_{ij} [[/math]]

The verification of the unitarity of [math]\bar{U}[/math] is similar. We first have:

[[math]] (\bar{U}U^t)_{ij} =\sum_{klm}u_{il}^*u_{jm}\otimes u_{lk}^*u_{mk} =\sum_{lm}u_{il}^*u_{jm}\otimes\delta_{lm} =\delta_{ij} [[/math]]

In the other sense the computation is similar, as follows:

[[math]] (U^t\bar{U})_{ij} =\sum_{klm}u_{kl}u_{km}^*\otimes u_{li}u_{mj}^* =\sum_{lm}\delta_{lm}\otimes u_{li}u_{mj}^* =\delta_{ij} [[/math]]

Regarding now the matrix [math]u^\varepsilon=1_N[/math], and also the matrix [math]u^S[/math], their biunitarity its clear. Thus, we can indeed define morphisms [math]\Delta,\varepsilon,S[/math] as in Definition 2.8, by using the universal properties of [math]C(O_N^+)[/math], [math]C(U_N^+)[/math], and this gives the result.

■

Let us study now the above quantum groups, with the techniques that we have. As a first observation, we have embeddings of compact quantum groups, as follows:

[[math]] \xymatrix@R=15mm@C=15mm{ U_N\ar[r]&U_N^+\\ O_N\ar[r]\ar[u]&O_N^+\ar[u] } [[/math]]

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, and from Proposition 2.14.

(2) This follows from the Gelfand theorem, which shows that we have presentation results for [math]C(O_N),C(U_N)[/math] as follows, similar to those in Theorem 2.23:

[[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]]

(3) This follows from (1) and from Proposition 2.11 above, 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].

■

As an interesting philosophical conclusion, if we denote by [math]L_N^+,F_N^+[/math] the discrete quantum groups which are dual to [math]O_N^+,U_N^+[/math], then we have embeddings as follows:

[[math]] L_N\subset L_N^+\quad,\quad F_N\subset F_N^+ [[/math]]

Thus [math]F_N^+[/math] is some kind of “free free group”, and [math]L_N^+[/math] is its real counterpart. This is not surprising, since [math]F_N,L_N[/math] are not “fully free”, their group algebras being cocommutative.

The last assertion in Theorem 2.24 suggests the following construction, from :

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 2.24 (3) tells us that the diagonal tori of [math]O_N^+,U_N^+[/math] are the group duals [math]\widehat{L}_N,\widehat{F}_N[/math]. We will be back to this later.

Here is now a more subtle result on [math]O_N^+,U_N^+[/math], having no classical counterpart:

Proposition

Consider the quantum groups [math]O_N^+,U_N^+[/math], with the corresponding fundamental corepresentations denoted [math]v,u[/math], and let [math]z=id\in C(\mathbb T)[/math].

• We have a morphism [math]C(U_N^+)\to C(\mathbb T)*C(O_N^+)[/math], given by [math]u=zv[/math].
• In other words, we have a quantum group embedding [math]\widetilde{O_N^+}\subset U_N^+[/math].
• This embedding is an isomorphism at the level of the diagonal tori.

Show Proof

The first two assertions follow from Proposition 2.19, or simply from the fact that [math]u=zv[/math] is biunitary. As for the third assertion, the idea here is that we have a similar model for the free group [math]F_N[/math], which is well-known to be faithful, [math]F_N\subset\mathbb Z*L_N[/math].

■

We will be back to the above morphism later on, with a proof of its faithfulness, after performing a suitable GNS construction, with respect to the Haar functionals.

Let us construct now some more examples of compact quantum groups. Following , , , , we can introduce some intermediate liberations, as follows:

Proposition

We have intermediate quantum groups as follows,

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

with [math]*[/math] standing for the fact that [math]u_{ij},u_{ij}^*[/math] must satisfy the relations [math]abc=cba[/math].

Show Proof

This is similar to the proof of Theorem 2.23, by using the elementary fact that if the entries of [math]u=(u_{ij})[/math] half-commute, then so do the entries of [math]u^\Delta[/math], [math]u^\varepsilon[/math], [math]u^S[/math].

■

In the same spirit, we have as well intermediate spheres as follows, with the symbol [math]*[/math] standing for the fact that [math]x_i,x_i^*[/math] must satisfy the relations [math]abc=cba[/math]:

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

At the level of the diagonal tori, we have the following result:

Theorem

The tori of the basic spheres and quantum groups are as follows,

[[math]] \xymatrix@R=15mm@C=15mm{ \widehat{\mathbb Z^N}\ar[r]&\widehat{\mathbb Z^{\circ N}}\ar[r]&\widehat{\mathbb Z^{*N}}\\ \widehat{\mathbb Z_2^N}\ar[r]\ar[u]&\widehat{\mathbb Z_2^{\circ N}}\ar[r]\ar[u]&\widehat{\mathbb Z_2^{*N}}\ar[u]} [[/math]]

with [math]\circ[/math] standing for the half-classical product operation for groups.

Show Proof

The idea here is as follows:

(1) The result on the left is well-known.

(2) The result on the right follows from Theorem 2.24 (3).

(3) The middle result follows as well, by imposing the relations [math]abc=cba[/math].

■

Let us discuss now the relation with the noncommutative spheres. Having the things started here is a bit tricky, and as a main source of inspiration, we have:

Proposition

Given an algebraic manifold [math]X\subset S^{N-1}_\mathbb C[/math], the formula

[[math]] G(X)=\left\{U\in U_N\Big|U(X)=X\right\} [[/math]]

defines a compact group of unitary matrices, or isometries, called affine isometry group of [math]X[/math]. For the spheres [math]S^{N-1}_\mathbb R,S^{N-1}_\mathbb C[/math] we obtain in this way the groups [math]O_N,U_N[/math].

Show Proof

The fact that [math]G(X)[/math] as defined above is indeed a group is clear, its compactness is clear as well, and finally the last assertion is clear as well. In fact, all this works for any closed subset [math]X\subset\mathbb C^N[/math], but we are not interested here in such general spaces.

■

We have the following quantum analogue of the above construction:

Proposition

Given an algebraic manifold [math]X\subset S^{N-1}_{\mathbb C,+}[/math], the category of the closed subgroups [math]G\subset U_N^+[/math] acting affinely on [math]X[/math], in the sense that the formula

[[math]] \Phi(x_i)=\sum_jx_j\otimes u_{ji} [[/math]]

defines a morphism of [math]C^*[/math]-algebras as follows,

[[math]] \Phi:C(X)\to C(X)\otimes C(G) [[/math]]

has a universal object, denoted [math]G^+(X)[/math], and called affine quantum isometry group of [math]X[/math].

Show Proof

Observe first that in the case where [math]\Phi[/math] as above exists, this morphism is automatically a coaction, in the sense that it satisfies the following conditions:

[[math]] (\Phi\otimes id)\Phi=(id\otimes\Delta)\Phi [[/math]]

[[math]] (id\otimes\varepsilon)\Phi=id [[/math]]

In order to prove now the result, assume that [math]X\subset S^{N-1}_{\mathbb C,+}[/math] comes as follows:

[[math]] C(X)=C(S^{N-1}_{\mathbb C,+})\Big/\Big \lt f_\alpha(x_1,\ldots,x_N)=0\Big \gt [[/math]]

Consider now the following variables:

[[math]] X_i=\sum_jx_j\otimes u_{ji}\in C(X)\otimes C(U_N^+) [[/math]]

Our claim is that [math]G=G^+(X)[/math] in the statement appears as follows:

[[math]] C(G)=C(U_N^+)\Big/\Big \lt f_\alpha(X_1,\ldots,X_N)=0\Big \gt [[/math]]

In order to prove this claim, we have to clarify how the relations [math]f_\alpha(X_1,\ldots,X_N)=0[/math] are interpreted inside [math]C(U_N^+)[/math], and then show that [math]G[/math] is indeed a quantum group. So, pick one of the defining polynomials, [math]f=f_\alpha[/math], and write it as follows:

[[math]] f(x_1,\ldots,x_N)=\sum_r\sum_{i_1^r\ldots i_{s_r}^r}\lambda_r\cdot x_{i_1^r}\ldots x_{i_{s_r}^r} [[/math]]

With [math]X_i=\sum_jx_j\otimes u_{ji}[/math] as above, we have the following formula:

[[math]] f(X_1,\ldots,X_N) =\sum_r\sum_{i_1^r\ldots i_{s_r}^r}\lambda_r\sum_{j_1^r\ldots j_{s_r}^r}x_{j_1^r}\ldots x_{j_{s_r}^r}\otimes u_{j_1^ri_1^r}\ldots u_{j_{s_r}^ri_{s_r}^r} [[/math]]

Since the variables on the right span a certain finite dimensional space, the relations [math]f(X_1,\ldots,X_N)=0[/math] correspond to certain relations between the variables [math]u_{ij}[/math]. Thus, we have indeed a closed subspace [math]G\subset U_N^+[/math], coming with a universal map:

[[math]] \Phi:C(X)\to C(X)\otimes C(G) [[/math]]

In order to show now that [math]G[/math] is a quantum group, consider the following elements:

[[math]] u_{ij}^\Delta=\sum_ku_{ik}\otimes u_{kj}\quad,\quad u_{ij}^\varepsilon=\delta_{ij}\quad,\quad u_{ij}^S=u_{ji}^* [[/math]]

Consider as well the following associated elements, with [math]\gamma\in\{\Delta,\varepsilon,S\}[/math]:

[[math]] X_i^\gamma=\sum_jx_j\otimes u_{ji}^\gamma [[/math]]

From the relations [math]f(X_1,\ldots,X_N)=0[/math] we deduce that we have:

[[math]] f(X_1^\gamma,\ldots,X_N^\gamma) =(id\otimes\gamma)f(X_1,\ldots,X_N) =0 [[/math]]

But this shows that for any exponent [math]\gamma\in\{\Delta,\varepsilon,S\}[/math] we can map [math]u_{ij}\to u_{ij}^\gamma[/math], and it follows that [math]G[/math] is indeed a compact quantum group, and we are done.

■

Following and related papers, we can now formulate:

Theorem

The quantum isometry groups of the basic spheres are

[[math]] \xymatrix@R=15mm@C=17mm{ U_N\ar[r]&U_N^*\ar[r]&U_N^+\\ O_N\ar[r]\ar[u]&O_N^*\ar[r]\ar[u]&O_N^+\ar[u]} [[/math]]

modulo identifying, as usual, the various [math]C^*[/math]-algebraic completions.

Show Proof

Let us first construct an action [math]U_N^+\curvearrowright S^{N-1}_{\mathbb C,+}[/math]. We must prove here that the variables [math]X_i=\sum_jx_j\otimes u_{ji}[/math] satisfy the defining relations for [math]S^{N-1}_{\mathbb C,+}[/math], namely:

[[math]] \sum_ix_ix_i^*=\sum_ix_i^*x_i=1 [[/math]]

But this follows from the biunitarity of [math]u[/math]. We have indeed:

[[math]] \begin{eqnarray*} \sum_iX_iX_i^* &=&\sum_{ijk}x_jx_k^*\otimes u_{ji}u_{ki}^*\\ &=&\sum_jx_jx_j^*\otimes1\\ &=&1\otimes1 \end{eqnarray*} [[/math]]

In the other sense the computation is similar, as follows:

[[math]] \begin{eqnarray*} \sum_iX_i^*X_i &=&\sum_{ijk}x_j^*x_k\otimes u_{ji}^*u_{ki}\\ &=&\sum_jx_j^*x_j\otimes1\\ &=&1\otimes1 \end{eqnarray*} [[/math]]

Regarding now [math]O_N^+\curvearrowright S^{N-1}_{\mathbb R,+}[/math], here we must check the extra relations [math]X_i=X_i^*[/math], and these are clear from [math]u_{ia}=u_{ia}^*[/math]. Finally, regarding the remaining actions, the verifications are clear as well, because if the coordinates [math]u_{ia}[/math] and [math]x_a[/math] are subject to commutation relations of type [math]ab=ba[/math], or of type [math]abc=cba[/math], then so are the variables [math]X_i=\sum_jx_j\otimes u_{ji}[/math].

We must prove now that all these actions are universal:

\underline{[math]S^{N-1}_{\mathbb R,+},S^{N-1}_{\mathbb C,+}[/math].} The universality of [math]U_N^+\curvearrowright S^{N-1}_{\mathbb C,+}[/math] is trivial by definition. As for the universality of [math]O_N^+\curvearrowright S^{N-1}_{\mathbb R,+}[/math], this comes from the fact that [math]X_i=X_i^*[/math], with [math]X_i=\sum_jx_j\otimes u_{ji}[/math] as above, gives [math]u_{ia}=u_{ia}^*[/math]. Thus [math]G\curvearrowright S^{N-1}_{\mathbb R,+}[/math] implies [math]G\subset O_N^+[/math], as desired.

\underline{[math]S^{N-1}_\mathbb R,S^{N-1}_\mathbb C[/math].} We use here a trick from Bhowmick-Goswami . Assuming first that we have an action [math]G\curvearrowright S^{N-1}_\mathbb R[/math], consider the following variables:

[[math]] w_{kl,ij}=u_{ki}u_{lj} [[/math]]

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

In terms of these variables, which can be thought of as being projective coordinates, the corresponding projective coaction map is given by:

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

We have the following formulae:

[[math]] \begin{eqnarray*} \Phi(p_{ij})&=&\sum_{k \lt l}p_{kl}\otimes(w_{kl,ij}+w_{lk,ij})+\sum_kp_{kk}\otimes w_{kk,ij}\\ \Phi(p_{ji})&=&\sum_{k \lt l}p_{kl}\otimes(w_{kl,ji}+w_{lk,ji})+\sum_kp_{kk}\otimes w_{kk,ji} \end{eqnarray*} [[/math]]

By comparing these two formulae, and then by using the linear independence of the variables [math]p_{kl}=x_kx_l[/math] with [math]k\leq l[/math], we conclude that we must have:

[[math]] w_{kl,ij}+w_{lk,ij}=w_{kl,ji}+w_{lk,ji} [[/math]]

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

[[math]] S(w_{kl,ij}) =S(u_{ki}u_{lj}) =S(u_{lj})S(u_{ki}) =u_{jl}u_{ik} =w_{ji,lk} [[/math]]

Thus by applying the antipode we obtain:

[[math]] w_{ji,lk}+w_{ji,kl}=w_{ij,lk}+w_{ij,kl} [[/math]]

By relabelling the indices, we obtain from this:

[[math]] w_{kl,ij}+w_{kl,ji}=w_{lk,ij}+w_{lk,ji} [[/math]]

Now by comparing with the original relation, we obtain:

[[math]] w_{lk,ij}=w_{kl,ji} [[/math]]

But, recalling that we have [math]w_{kl,ij}=u_{ki}u_{lj}[/math], this formula reads:

[[math]] u_{li}u_{kj}=u_{kj}u_{li} [[/math]]

We therefore conclude we have [math]G\subset O_N[/math], as claimed. The proof of the universality of the action [math]U_N\curvearrowright S^{N-1}_\mathbb C[/math] is similar.

\underline{[math]S^{N-1}_{\mathbb R,*},S^{N-1}_{\mathbb C,*}[/math].} Assume that we have an action [math]G\curvearrowright S^{N-1}_{\mathbb C,*}[/math]. From [math]\Phi(x_a)=\sum_ix_i\otimes u_{ia}[/math] we obtain then that, with [math]p_{ab}=z_a\bar{z}_b[/math], we have:

[[math]] \Phi(p_{ab})=\sum_{ij}p_{ij}\otimes u_{ia}u_{jb}^* [[/math]]

By multiplying these two formulae, we obtain:

[[math]] \begin{eqnarray*} \Phi(p_{ab}p_{cd})&=&\sum_{ijkl}p_{ij}p_{kl}\otimes u_{ia}u_{jb}^*u_{kc}u_{ld}^*\\ \Phi(p_{ad}p_{cb})&=&\sum_{ijkl}p_{il}p_{kj}\otimes u_{ia}u_{ld}^*u_{kc}u_{jb}^* \end{eqnarray*} [[/math]]

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

[[math]] \sum_{ijkl}p_{ij}p_{kl}\otimes u_{ia}[u_{jb}^*,u_{kc},u_{ld}^*]=0 [[/math]]

Since the variables [math]p_{ij}p_{kl}=z_i\bar{z}_jz_k\bar{z}_l[/math] depend only on [math]|\{i,k\}|,|\{j,l\}|\in\{1,2\}[/math], and this dependence produces the only relations between them, we are led to [math]4[/math] equations:

(1) [math]u_{ia}[u_{jb}^*,u_{ka},u_{lb}^*]=0[/math], [math]\forall a,b[/math].

(2) [math]u_{ia}[u_{jb}^*,u_{ka},u_{ld}^*]+u_{ia}[u_{jd}^*,u_{ka},u_{lb}^*]=0[/math], [math]\forall a[/math], [math]\forall b\neq d[/math].

(3) [math]u_{ia}[u_{jb}^*,u_{kc},u_{lb}^*]+u_{ic}[u_{jb}^*,u_{ka},u_{lb}^*]=0[/math], [math]\forall a\neq c[/math], [math]\forall b[/math].

(4) [math]u_{ia}([u_{jb}^*,u_{kc},u_{ld}^*]+[u_{jd}^*,u_{kc},u_{lb}^*])+u_{ic}([u_{jb}^*,u_{ka},u_{ld}^*]+[u_{jd}^*,u_{ka},u_{lb}^*])=0,\forall a\neq c,b\neq d[/math].

From (1,2) we conclude that (2) holds with no restriction on the indices. By multiplying now this formula to the left by [math]u_{ia}^*[/math], and then summing over [math]i[/math], we obtain:

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

By applying now the antipode, then the involution, and finally by suitably relabelling all the indices, we successively obtain from this formula:

[[math]] \begin{eqnarray*} &&[u_{dl},u_{ak}^*,u_{bj}]+[u_{bl},u_{ak}^*,u_{dj}]=0\\ &\implies&[u_{dl}^*,u_{ak},u_{bj}^*]+[u_{bl}^*,u_{ak},u_{dj}^*]=0\\ &\implies&[u_{ld}^*,u_{ka},u_{jb}^*]+[u_{jd}^*,u_{ka},u_{lb}^*]=0 \end{eqnarray*} [[/math]]

Now by comparing with the original relation, above, we conclude that we have:

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

Thus we have reached to the formulae defining [math]U_N^*[/math], and we are done. Finally, in what regards the universality of the action [math]O_N^*\curvearrowright S^{N-1}_{\mathbb R,*}[/math], this follows from the universality of the actions [math]U_N^*\curvearrowright S^{N-1}_{\mathbb C,*}[/math] and of [math]O_N^+\curvearrowright S^{N-1}_{\mathbb R,+}[/math], and from [math]U_N^*\cap O_N^+=O_N^*[/math].

■

As a conclusion to all this, we have now a simple and reliable definition for the compact quantum groups, in the Lie case, namely [math]G\subset U_N^+[/math], covering all the compact Lie groups, [math]G\subset U_N[/math], covering as well all the duals [math]\widehat{\Gamma}[/math] of the finitely generated groups, [math]F_N\to\Gamma[/math], and allowing the construction of several interesting examples, such as [math]O_N^+,U_N^+[/math].

With respect to the noncommutative geometry questions raised in chapter 1 above, we certainly have here some advances. In order to further advance, however, we would need now representation theory results, in the spirit of Weyl , for our quantum isometry groups. We will develop all this in what follows, in the next few chapters.

General references

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