2a. Quantum groups

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

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We have seen so far that the pairs sphere/torus [math](S,T)[/math] corresponding to the real and complex geometries, of [math]\mathbb R^N,\mathbb C^N[/math], have some natural free analogues. Our objective now will be that of adding to the picture a pair of quantum groups [math](U,K)[/math], as to reach to a quadruplet of objects [math](S,T,U,K)[/math], with relations between them, as follows:

[[math]] \xymatrix@R=50pt@C=50pt{ S\ar[r]\ar[d]\ar[dr]&T\ar[l]\ar[d]\ar[dl]\\ U\ar[u]\ar[ur]\ar[r]&K\ar[l]\ar[ul]\ar[u] } [[/math]]

Before starting, some philosophical comments. You might argue that the pairs [math](S,T)[/math] that we have look just fine, so why embarking into quantum groups, and complicating our theory with objects [math](U,K)[/math]. This is a reasonable criticism, and in answer:

(1) First of all, there is no sphere [math]S[/math] without corresponding rotation group [math]U[/math]. With this meaning that, no matter what you want to do with [math]S[/math], of reasonably advanced type, like integrating over it, or looking at its Laplacian, and so on, you will certainly run into [math]U[/math]. And for similar reasons, a bit more complicated, there is no [math]T[/math] without [math]K[/math] either.

(2) This being said, you will say, why not further studying [math]S,T[/math], say from a differential geometry viewpoint, and leaving [math]U,K[/math] for later. Well, this does not work. The problem is that [math]S,T[/math], at least in the free case, that we are very interested in here, while having a Laplacian, do not have a Dirac operator in the sense of Connes .

(3) In short, such ideas will not work, and we are led into [math]U,K[/math]. By the way, meditating a bit about noncommutative differential geometry, at this point, is something recommended. And that you will have to do by yourself, the no-go results here being folklore. The needed read here is Connes , with Blackadar helping.

So, quantum groups. We will spend quite some time in introducing them, and working out their properties, and with this long series of things to be learned being good news, because the more theory we have about quantum groups, the more techniques we will have for dealing with [math](U,K)[/math], and so with the whole quadruplets [math](S,T,U,K)[/math].

The formalism that we need, coming from Woronowicz , is as follows:

Definition

A Woronowicz algebra is a [math]C^*[/math]-algebra [math]A[/math], given with a unitary matrix [math]u\in M_N(A)[/math] whose coefficients generate [math]A[/math], such that the formulae

[[math]] \Delta(u_{ij})=\sum_ku_{ik}\otimes u_{kj} [[/math]]

[[math]] \varepsilon(u_{ij})=\delta_{ij} [[/math]]

[[math]] S(u_{ij})=u_{ji}^* [[/math]]

define morphisms of [math]C^*[/math]-algebras as follows,

[[math]] \Delta:A\to A\otimes A [[/math]]

[[math]] \varepsilon:A\to\mathbb C [[/math]]

[[math]] S:A\to A^{opp} [[/math]]

called comultiplication, counit and antipode.

Obviously, this is something tricky, and we will see details in a moment, the idea being that these are the axioms which best fit with what we want to do, in this book. Let us also mention, technically, that [math]\otimes[/math] in the above can be any topological tensor product, and with the choice of [math]\otimes[/math] being irrelevant, but more on this later. Also, [math]A^{opp}[/math] is the opposite algebra, with multiplication [math]a\cdot b=ba[/math], and more on this later too.

We say that [math]A[/math] is cocommutative when [math]\Sigma\Delta=\Delta[/math], where [math]\Sigma(a\otimes b)=b\otimes a[/math] is the flip. With this convention, we have the following key result, from Woronowicz :

Theorem

The following are Woronowicz algebras:

• [math]C(G)[/math], with [math]G\subset U_N[/math] compact Lie group. Here the structural maps are:
  [[math]] \Delta(\varphi)=(g,h)\to \varphi(gh) [[/math]]
  [[math]] \varepsilon(\varphi)=\varphi(1) [[/math]]
  [[math]] S(\varphi)=g\to\varphi(g^{-1}) [[/math]]
• [math]C^*(\Gamma)[/math], with [math]F_N\to\Gamma[/math] finitely generated group. Here the structural maps are:
  [[math]] \Delta(g)=g\otimes g [[/math]]
  [[math]] \varepsilon(g)=1 [[/math]]
  [[math]] S(g)=g^{-1} [[/math]]

Moreover, we obtain in this way all the commutative/cocommutative algebras.

Show Proof

In both cases, we have to exhibit a certain matrix [math]u[/math]:

(1) Here we can use the matrix [math]u=(u_{ij})[/math] formed by matrix coordinates of [math]G[/math]:

[[math]] g=\begin{pmatrix} u_{11}(g)&\ldots&u_{1N}(g)\\ \vdots&&\vdots\\ u_{N1}(g)&\ldots&u_{NN}(g) \end{pmatrix} [[/math]]

(2) Here we can use the diagonal matrix formed by generators of [math]\Gamma[/math]:

[[math]] u=\begin{pmatrix} g_1&&0\\ &\ddots&\\ 0&&g_N \end{pmatrix} [[/math]]

Finally, the last assertion follows from the Gelfand theorem, in the commutative case. In the cocommutative case, this is something more technical, to be discussed later.

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In general now, the structural maps [math]\Delta,\varepsilon,S[/math] have the following properties:

Proposition

Let [math](A,u)[/math] be a Woronowicz algebra.

• [math]\Delta,\varepsilon[/math] satisfy the usual axioms for a comultiplication and a counit, namely:
  [[math]] (\Delta\otimes id)\Delta=(id\otimes \Delta)\Delta [[/math]]
  [[math]] (\varepsilon\otimes id)\Delta=(id\otimes\varepsilon)\Delta=id [[/math]]
• [math]S[/math] satisfies the antipode axiom, on the [math]*[/math]-subalgebra generated by entries of [math]u[/math]:
  [[math]] m(S\otimes id)\Delta=m(id\otimes S)\Delta=\varepsilon(.)1 [[/math]]
• In addition, the square of the antipode is the identity, [math]S^2=id[/math].

Show Proof

Observe first that the result holds in the case where [math]A[/math] is commutative. Indeed, by using Theorem 2.2 (1) we can write:

[[math]] \Delta=m^T [[/math]]

[[math]] \varepsilon=u^T [[/math]]

[[math]] S=i^T [[/math]]

The 3 conditions in the statement come then by transposition from the basic 3 group theory conditions satisfied by [math]m,u,i[/math], namely:

[[math]] m(m\times id)=m(id\times m) [[/math]]

[[math]] m(id\times u)=m(u\times id)=id [[/math]]

[[math]] m(id\times i)\delta=m(i\times id)\delta=1 [[/math]]

Here [math]\delta(g)=(g,g)[/math]. Observe also that the last condition, [math]S^2=id[/math], is satisfied as well, coming from the identity [math]i^2=id[/math], which is a consequence of the group axioms.

Observe also that the result holds as well in the case where [math]A[/math] is cocommutative, by using Theorem 2.2 (1). Indeed, the 3 formulae in the statement are all trivial, and the condition [math]S^2=id[/math] follows once again from the group theory formula [math](g^{-1})^{-1}=g[/math].

In the general case now, the proof goes as follows:

(1) We have the following computation:

[[math]] (\Delta\otimes id)\Delta(u_{ij}) =\sum_l\Delta(u_{il})\otimes u_{lj} =\sum_{kl}u_{ik}\otimes u_{kl}\otimes u_{lj} [[/math]]

We have as well the following computation, which gives the first formula:

[[math]] (id\otimes\Delta)\Delta(u_{ij}) =\sum_ku_{ik}\otimes\Delta(u_{kj}) =\sum_{kl}u_{ik}\otimes u_{kl}\otimes u_{lj} [[/math]]

On the other hand, we have the following computation:

[[math]] (id\otimes\varepsilon)\Delta(u_{ij}) =\sum_ku_{ik}\otimes\varepsilon(u_{kj}) =u_{ij} [[/math]]

We have as well the following computation, which gives the second formula:

[[math]] (\varepsilon\otimes id)\Delta(u_{ij}) =\sum_k\varepsilon(u_{ik})\otimes u_{kj} =u_{ij} [[/math]]

(2) By using the fact that the matrix [math]u=(u_{ij})[/math] is unitary, we obtain:

[[math]] \begin{eqnarray*} m(id\otimes S)\Delta(u_{ij}) &=&\sum_ku_{ik}S(u_{kj})\\ &=&\sum_ku_{ik}u_{jk}^*\\ &=&(uu^*)_{ij}\\ &=&\delta_{ij} \end{eqnarray*} [[/math]]

We have as well the following computation, which gives the result:

[[math]] \begin{eqnarray*} m(S\otimes id)\Delta(u_{ij}) &=&\sum_kS(u_{ik})u_{kj}\\ &=&\sum_ku_{ki}^*u_{kj}\\ &=&(u^*u)_{ij}\\ &=&\delta_{ij} \end{eqnarray*} [[/math]]

(3) Finally, the formula [math]S^2=id[/math] holds as well on the generators, and we are done.

■

Let us record as well the following technical result:

Proposition

Given a Woronowicz algebra [math](A,u)[/math], we have [math]u^t=\bar{u}^{-1}[/math], so [math]u[/math] is biunitary, in the sense that it is unitary, with unitary transpose.

Show Proof

We have the following computation, based on the fact that [math]u[/math] is unitary:

[[math]] \begin{eqnarray*} (uu^*)_{ij}=\delta_{ij} &\implies&\sum_kS(u_{ik}u_{jk}^*)=\delta_{ij}\\ &\implies&\sum_ku_{kj}u_{ki}^*=\delta_{ij}\\ &\implies&(u^t\bar{u})_{ji}=\delta_{ij} \end{eqnarray*} [[/math]]

Similarly, we have the following computation, once agan using the unitarity of [math]u[/math]:

[[math]] \begin{eqnarray*} (u^*u)_{ij}=\delta_{ij} &\implies&\sum_kS(u_{ki}^*u_{kj})=\delta_{ij}\\ &\implies&\sum_ku_{jk}^*u_{ik}=\delta_{ij}\\ &\implies&(\bar{u}u^t)_{ji}=\delta_{ij} \end{eqnarray*} [[/math]]

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

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Summarizing, the Woronowicz algebras appear to have nice properties. In view of Theorem 2.2 and of Proposition 2.3, we can formulate the following definition:

Definition

Given a Woronowicz algebra [math]A[/math], we formally write

[[math]] A=C(G)=C^*(\Gamma) [[/math]]

and call [math]G[/math] compact quantum group, and [math]\Gamma[/math] discrete quantum group.

When [math]A[/math] is commutative and cocommutative, [math]G[/math] and [math]\Gamma[/math] are usual abelian groups, dual to each other. In general, we still agree to write [math]G=\widehat{\Gamma},\Gamma=\widehat{G}[/math], but in a formal sense. As a final piece of general theory now, let us complement Definition 2.1 with:

Definition

Given two Woronowicz algebras [math](A,u)[/math] and [math](B,v)[/math], we write

[[math]] A\simeq B [[/math]]

and identify the corresponding quantum groups, when we have an isomorphism

[[math]] \lt u_{ij} \gt \simeq \lt v_{ij} \gt [[/math]]

of [math]*[/math]-algebras, mapping standard coordinates to standard coordinates.

With this convention, which is in tune with our conventions for algebraic manifolds from chapter 1, and more on this later, any compact or discrete quantum group corresponds to a unique Woronowicz algebra, up to equivalence. Also, we can see now why in Definition 2.1 the choice of the exact topological tensor product [math]\otimes[/math] is irrelevant. Indeed, no matter what tensor product [math]\otimes[/math] we use there, we end up with the same Woronowicz algebra, and the same compact and discrete quantum groups, up to equivalence.

In practice, we will use in what follows the simplest such tensor product [math]\otimes[/math], which is the maximal one, obtained as completion of the usual algebraic tensor product with respect to the biggest [math]C^*[/math]-norm. With the remark that this product is something rather abstract, and so can be treated, in practice, as a usual algebraic tensor product.

Going ahead now, let us call corepresentation of [math]A[/math] any unitary matrix [math]v\in M_n(\mathcal A)[/math], where [math]\mathcal A= \lt u_{ij} \gt [/math], satisfying the same conditions are those satisfied by [math]u[/math], namely:

[[math]] \Delta(v_{ij})=\sum_kv_{ik}\otimes v_{kj} [[/math]]

[[math]] \varepsilon(v_{ij})=\delta_{ij} [[/math]]

[[math]] S(v_{ij})=v_{ji}^* [[/math]]

These corepresentations can be thought of as corresponding to the finite dimensional unitary smooth representations of the underlying compact quantum group [math]G[/math]. Following Woronowicz , we have the following key result:

Theorem

Any Woronowicz algebra [math]A=C(G)[/math] has a Haar integration functional,

[[math]] \left(\int_G\otimes id\right)\Delta=\left(id\otimes\int_G\right)\Delta=\int_G(.)1 [[/math]]

which can be constructed by starting with any faithful positive form [math]\varphi\in A^*[/math], and setting

[[math]] \int_G=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\varphi^{*k} [[/math]]

where [math]\phi*\psi=(\phi\otimes\psi)\Delta[/math]. Moreover, for any corepresentation [math]v\in M_n(\mathbb C)\otimes A[/math] we have

[[math]] \left(id\otimes\int_G\right)v=P [[/math]]

where [math]P[/math] is the orthogonal projection onto [math]Fix(v)=\big\{\xi\in\mathbb C^n\big|v\xi=\xi\big\}[/math].

Show Proof

Following , this can be done in 3 steps, as follows:

(1) Given [math]\varphi\in A^*[/math], our claim is that the following limit converges, for any [math]a\in A[/math]:

[[math]] \int_\varphi a=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\varphi^{*k}(a) [[/math]]

Indeed, we can assume, by linearity, that [math]a[/math] is the coefficient of a corepresentation:

[[math]] a=(\tau\otimes id)v [[/math]]

But in this case, an elementary computation shows that we have the following formula, where [math]P_\varphi[/math] is the orthogonal projection onto the [math]1[/math]-eigenspace of [math](id\otimes\varphi)v[/math]:

[[math]] \left(id\otimes\int_\varphi\right)v=P_\varphi [[/math]]

(2) Since [math]v\xi=\xi[/math] implies [math][(id\otimes\varphi)v]\xi=\xi[/math], we have [math]P_\varphi\geq P[/math], where [math]P[/math] is the orthogonal projection onto the following fixed point space:

[[math]] Fix(v)=\left\{\xi\in\mathbb C^n\Big|v\xi=\xi\right\} [[/math]]

The point now is that when [math]\varphi\in A^*[/math] is faithful, by using a standard positivity trick, one can prove that we have [math]P_\varphi=P[/math]. Assume indeed [math]P_\varphi\xi=\xi[/math], and let us set:

[[math]] a=\sum_i\left(\sum_jv_{ij}\xi_j-\xi_i\right)\left(\sum_kv_{ik}\xi_k-\xi_i\right)^* [[/math]]

We must prove that we have [math]a=0[/math]. Since [math]v[/math] is biunitary, we have:

[[math]] \begin{eqnarray*} a &=&\sum_i\left(\sum_j\left(v_{ij}\xi_j-\frac{1}{N}\xi_i\right)\right)\left(\sum_k\left(v_{ik}^*\bar{\xi}_k-\frac{1}{N}\bar{\xi}_i\right)\right)\\ &=&\sum_{ijk}v_{ij}v_{ik}^*\xi_j\bar{\xi}_k-\frac{1}{N}v_{ij}\xi_j\bar{\xi}_i-\frac{1}{N}v_{ik}^*\xi_i\bar{\xi}_k+\frac{1}{N^2}\xi_i\bar{\xi}_i\\ &=&\sum_j|\xi_j|^2-\sum_{ij}v_{ij}\xi_j\bar{\xi}_i-\sum_{ik}v_{ik}^*\xi_i\bar{\xi}_k+\sum_i|\xi_i|^2\\ &=&||\xi||^2- \lt v\xi,\xi \gt -\overline{ \lt v\xi,\xi \gt }+||\xi||^2\\ &=&2(||\xi||^2-Re( \lt v\xi,\xi \gt )) \end{eqnarray*} [[/math]]

By using now our assumption [math]P_\varphi\xi=\xi[/math], we obtain from this:

[[math]] \begin{eqnarray*} \varphi(a) &=&2\varphi(||\xi||^2-Re( \lt v\xi,\xi \gt ))\\ &=&2(||\xi||^2-Re( \lt P_\varphi\xi,\xi \gt ))\\ &=&2(||\xi||^2-||\xi||^2)\\ &=&0 \end{eqnarray*} [[/math]]

Now since [math]\varphi[/math] is faithful, this gives [math]a=0[/math], and so [math]v\xi=\xi[/math]. Thus [math]\int_\varphi[/math] is independent of [math]\varphi[/math], and is given on coefficients [math]a=(\tau\otimes id)v[/math] by the following formula:

[[math]] \left(id\otimes\int_\varphi\right)v=P [[/math]]

(3) With the above formula in hand, the left and right invariance of [math]\int_G=\int_\varphi[/math] is clear on coefficients, and so in general, and this gives all the assertions. See .

■

Consider the dense [math]*[/math]-subalgebra [math]\mathcal A\subset A[/math] generated by the coefficients of the fundamental corepresentation [math]u[/math], and endow it with the following scalar product:

[[math]] \lt a,b \gt =\int_Gab^* [[/math]]

Once again following Woronowicz , we have the following result:

Theorem

We have the following Peter-Weyl type results:

• Any corepresentation decomposes as a sum of irreducible corepresentations.
• Each irreducible corepresentation appears inside a certain [math]u^{\otimes k}[/math].
• [math]\mathcal A=\bigoplus_{v\in Irr(A)}M_{\dim(v)}(\mathbb C)[/math], the summands being pairwise orthogonal.
• The characters of irreducible corepresentations form an orthonormal system.

Show Proof

All these results are from , the idea being as follows:

(1) Given a corepresentation [math]v\in M_n(A)[/math], consider its interwiner algebra:

[[math]] End(v)=\left\{T\in M_n(\mathbb C)\Big|Tv=vT\right\} [[/math]]

It is elementary to see that this is a finite dimensional [math]C^*[/math]-algebra, and we conclude from this that we have a decomposition as follows:

[[math]] End(v)=M_{n_1}(\mathbb C)\oplus\ldots\oplus M_{n_k}(\mathbb C) [[/math]]

To be more precise, such a decomposition appears by writing the unit of our algebra as a sum of minimal projections, as follows, and then working out the details:

[[math]] 1=p_1+\ldots+p_k [[/math]]

But this decomposition allows us to define subcorepresentations [math]v_i\subset v[/math], which are irreducible, so we obtain, as desired, a decomposition [math]v=v_1+\ldots+v_k[/math].

(2) To any corepresentation [math]v\in M_n(A)[/math] we associate its space of coefficients, given by [math]C(v)=span(v_{ij})[/math]. The construction [math]v\to C(v)[/math] is then functorial, in the sense that it maps subcorepresentations into subspaces. Observe also that we have:

[[math]] \mathcal A=\sum_{k\in\mathbb N*\mathbb N}C(u^{\otimes k}) [[/math]]

Now given an arbitrary corepresentation [math]v\in M_n(A)[/math], the corresponding coefficient space is a finite dimensional subspace [math]C(v)\subset\mathcal A[/math], and so we must have, for certain positive integers [math]k_1,\ldots,k_p[/math], an inclusion of vector spaces, as follows:

[[math]] C(v)\subset C(u^{\otimes k_1}\oplus\ldots\oplus u^{\otimes k_p}) [[/math]]

We deduce from this that we have an inclusion of corepresentations, as follows:

[[math]] v\subset u^{\otimes k_1}\oplus\ldots\oplus u^{\otimes k_p} [[/math]]

Thus, by using (1), we are led to the conclusion in the statement.

(3) By using (1) and (2), we obtain a linear space decomposition as follows:

[[math]] \mathcal A =\sum_{v\in Irr(A)}C(v) =\sum_{v\in Irr(A)}M_{\dim(v)}(\mathbb C) [[/math]]

In order to conclude, it is enough to prove that for any two irreducible corepresentations [math]v,w\in Irr(A)[/math], the corresponding spaces of coefficients are orthogonal:

[[math]] v\not\sim w\implies C(v)\perp C(w) [[/math]]

As a first observation, which follows from an elementary computation, for any two corepresentations [math]v,w[/math] we have a Frobenius type isomorphism, as follows:

[[math]] Hom(v,w)\simeq Fix(\bar{v}\otimes w) [[/math]]

Now let us set [math]P_{ia,jb}=\int_Gv_{ij}w_{ab}^*[/math]. According to Theorem 2.7, the matrix [math]P[/math] is the orthogonal projection onto the following vector space:

[[math]] Fix(v\otimes\bar{w}) \simeq Hom(\bar{v},\bar{w}) =\{0\} [[/math]]

Thus we have [math]P=0[/math], and so [math]C(v)\perp C(w)[/math], which gives the result.

(4) The algebra [math]\mathcal A_{central}[/math] contains indeed all the characters, because we have:

[[math]] \Sigma\Delta(\chi_v) =\sum_{ij}v_{ji}\otimes v_{ij} =\Delta(\chi_v) [[/math]]

The fact that the characters span [math]\mathcal A_{central}[/math], and form an orthogonal basis of it, follow from (3). Finally, regarding the norm 1 assertion, consider the following integrals:

[[math]] P_{ik,jl}=\int_Gv_{ij}v_{kl}^* [[/math]]

We know from Theorem 2.7 that these integrals form the orthogonal projection onto [math]Fix(v\otimes\bar{v})\simeq End(\bar{v})=\mathbb C1[/math]. By using this fact, we obtain the following formula:

[[math]] \int_G\chi_v\chi_v^* =\sum_{ij}\int_Gv_{ii}v_{jj}^* =\sum_i\frac{1}{N} =1 [[/math]]

Thus the characters have indeed norm 1, and we are done.

■

We refer to Woronowicz for full details on all the above, and for some applications as well. Let us just record here the fact that in the cocommutative case, we obtain from (4) that the irreducible corepresentations must be all 1-dimensional, and so that we must have [math]A=C^*(\Gamma)[/math] for some discrete group [math]\Gamma[/math], as mentioned in Theorem 2.2.

At a more technical level now, we have a number of more advanced results, from Woronowicz , and other papers, that must be known as well. We will present them quickly, and for details you check my book . First we have:

Theorem

Let [math]A_{full}[/math] be the enveloping [math]C^*[/math]-algebra of [math]\mathcal A[/math], and let [math]A_{red}[/math] be the quotient of [math]A[/math] by the null ideal of the Haar integration. The following are then equivalent:

• The Haar functional of [math]A_{full}[/math] is faithful.
• The projection map [math]A_{full}\to A_{red}[/math] is an isomorphism.
• The counit map [math]\varepsilon:A_{full}\to\mathbb C[/math] factorizes through [math]A_{red}[/math].
• We have [math]N\in\sigma(Re(\chi_u))[/math], the spectrum being taken inside [math]A_{red}[/math].

If this is the case, we say that the underlying discrete quantum group [math]\Gamma[/math] is amenable.

Show Proof

This is well-known in the group dual case, [math]A=C^*(\Gamma)[/math], with [math]\Gamma[/math] being a usual discrete group. In general, the result follows by adapting the group dual case proof:

[math](1)\iff(2)[/math] This simply follows from the fact that the GNS construction for the algebra [math]A_{full}[/math] with respect to the Haar functional produces the algebra [math]A_{red}[/math].

[math](2)\iff(3)[/math] Here [math]\implies[/math] is trivial, and conversely, a counit map [math]\varepsilon:A_{red}\to\mathbb C[/math] produces an isomorphism [math]A_{red}\to A_{full}[/math], via a formula of type [math](\varepsilon\otimes id)\Phi[/math].

[math](3)\iff(4)[/math] Here [math]\implies[/math] is clear, coming from [math]\varepsilon(N-Re(\chi (u)))=0[/math], and the converse can be proved by doing some standard functional analysis.

■

Yet another important result, also about the general Woronowicz algebras, and that we will be heavily using in what follows, is Tannakian duality, as follows:

Theorem

The following operations are inverse to each other:

• The construction [math]A\to C[/math], which associates to any Woronowicz algebra [math]A[/math] the tensor category formed by the intertwiner spaces [math]C_{kl}=Hom(u^{\otimes k},u^{\otimes l})[/math].
• The construction [math]C\to A[/math], which associates to a tensor category [math]C[/math] the Woronowicz algebra [math]A[/math] presented by the relations [math]T\in Hom(u^{\otimes k},u^{\otimes l})[/math], with [math]T\in C_{kl}[/math].

Show Proof

This is something quite deep, going back to Woronowicz in a slightly different form, and to Malacarne in the simplified form presented above. The idea is that this can be proved by doing some abstract algebra, as follows:

(1) We have indeed a construction [math]A\to C[/math] as above, whose output is a tensor [math]C^*[/math]-subcategory with duals of the tensor [math]C^*[/math]-category of Hilbert spaces.

(2) We have as well a construction [math]C\to A[/math] as above, simply by dividing the free [math]*[/math]-algebra on [math]N^2[/math] variables by the relations in the statement.

Regarding now the bijection claim, some elementary algebra shows that [math]C=C_{A_C}[/math] implies [math]A=A_{C_A}[/math], and also that [math]C\subset C_{A_C}[/math] is automatic. Thus we are left with proving [math]C_{A_C}\subset C[/math]. But this latter inclusion can be proved indeed, by doing some algebra, and using von Neumann's bicommutant theorem, in finite dimensions. See .

■

As a concrete consequence of the above result, we have:

Theorem

We have an embedding as follows, using double indices,

[[math]] G\subset S^{N^2-1}_{\mathbb C,+}\quad,\quad x_{ij}=\frac{u_{ij}}{\sqrt{N}} [[/math]]

making [math]G[/math] an algebraic submanifold of the free sphere.

Show Proof

The fact that we have an embedding as above follows from the fact that [math]u=(u_{ij})[/math] is biunitary, that we know from Proposition 2.4. As for the algebricity claim, this follows from Theorem 2.10. Indeed, assuming that [math]A=C(G)[/math] is of the form [math]A=A_C[/math], it follows that [math]G[/math] is algebraic. But this is always the case, because we can take [math]C=C_A[/math].

■

Observe that the embedding constructed above makes the link between our isomorphim conventions for quantum groups and for algebraic manifolds.

General references

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