7c. Quantum groups

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The duals of discrete groups have several similarities with the compact groups, and our goal now will be that of unifying these two classes of compact quantum spaces. Let us start with the following definition, due to Woronowicz :

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}\quad,\quad \varepsilon(u_{ij})=\delta_{ij}\quad,\quad S(u_{ij})=u_{ji}^* [[/math]]

define morphisms of [math]C^*[/math]-algebras [math]\Delta:A\to A\otimes A[/math], [math]\varepsilon:A\to\mathbb C[/math], [math]S:A\to A^{opp}[/math].

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. We have the following result, which justifies the terminology and axioms:

Proposition

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)\quad,\quad \varepsilon(\varphi)=\varphi(1)\quad,\quad 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\quad,\quad \varepsilon(g)=1\quad,\quad 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]. For the first assertion, we can use the matrix [math]u=(u_{ij})[/math] formed by matrix coordinates of [math]G[/math], given by:

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

As for the second assertion, here we can use the diagonal matrix formed by generators, [math]u=diag(g_1,\ldots,g_N)[/math]. Finally, the last assertion follows from the Gelfand theorem, in the commutative case, and in the cocommutative case, we will be back to this 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

When [math]A[/math] is commutative, by using Proposition 7.26 we can write:

[[math]] \Delta=m^t\quad,\quad \varepsilon=u^t\quad,\quad S=i^t [[/math]]

The above 3 conditions come then by transposition from the basic 3 group theory conditions satisfied by [math]m,u,i[/math], which are as follows, with [math]\delta(g)=(g,g)[/math]:

[[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]]

Observe that [math]S^2=id[/math] is satisfied as well, coming from [math]i^2=id[/math]. In general now, all the formulae in the statement are satisfied on the generators [math]u_{ij}[/math], and so by linearity, multiplicativity and continuity they are satisfied everywhere, as desired.

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In view of Proposition 7.26, 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 both commutative and cocommutative, [math]G[/math] is a compact abelian group, [math]\Gamma[/math] is a discrete abelian group, and these groups are dual to each other:

[[math]] G=\widehat{\Gamma}\quad,\quad \Gamma=\widehat{G} [[/math]]

In general, we still agree to write [math]G=\widehat{\Gamma},\Gamma=\widehat{G}[/math], in a formal sense. Finally, in relation with functoriality issues, let us complement Definitions 7.25 and 7.28 with:

Definition

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

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

and we identify as well the corresponding compact and discrete quantum groups, when we have an isomorphism of [math]*[/math]-algebras [math] \lt u_{ij} \gt \simeq \lt v_{ij} \gt [/math], mapping [math]u_{ij}\to v_{ij}[/math].

In order to develop now some theory, let us call corepresentation of [math]A[/math] any unitary matrix [math]v\in M_n(\mathcal A)[/math], with [math]\mathcal A= \lt u_{ij} \gt [/math], satisfying the same conditions as [math]u[/math], namely:

[[math]] \Delta(v_{ij})=\sum_kv_{ik}\otimes v_{kj}\quad,\quad \varepsilon(v_{ij})=\delta_{ij}\quad,\quad S(v_{ij})=v_{ji}^* [[/math]]

These can be thought of as corresponding to the unitary representations of the underlying compact quantum group [math]G[/math]. Following Woronowicz , we have:

Theorem

Any Woronowicz algebra has a unique 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)=\{\xi\in\mathbb C^n|v\xi=\xi\}[/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, by linearity we can assume that [math]a[/math] is the coefficient of 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 space [math]Fix(v)=\{\xi\in\mathbb C^n|v\xi=\xi\}[/math]. The point now is that when [math]\varphi\in A^*[/math] is faithful, by using a positivity trick, one can prove that we have [math]P_\varphi=P[/math]. Thus our linear form [math]\int_\varphi[/math] is independent of [math]\varphi[/math], and is given on coefficients [math]a=(\tau\otimes id)v[/math] by:

[[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 .

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As a main application, we can develop a Peter-Weyl type theory for the corepresentations of [math]A[/math]. 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]]

With this convention, we have the following result, also from Woronowicz :

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 [math]v\in M_n(A)[/math], its intertwiner algebra [math]End(v)=\{T\in M_n(\mathbb C)|Tv=vT\}[/math] is a finite dimensional [math]C^*[/math]-algebra, and so decomposes as [math]End(v)=M_{n_1}(\mathbb C)\oplus\ldots\oplus M_{n_r}(\mathbb C)[/math]. But this gives a decomposition of type [math]v=v_1+\ldots+v_r[/math], as desired.

(2) Consider indeed the Peter-Weyl corepresentations, [math]u^{\otimes k}[/math] with [math]k[/math] colored integer, defined by [math]u^{\otimes\emptyset}=1[/math], [math]u^{\otimes\circ}=u[/math], [math]u^{\otimes\bullet}=\bar{u}[/math] and multiplicativity. The coefficients of these corepresentations span the dense algebra [math]\mathcal A[/math], and by using (1), this gives the result.

(3) Here the direct sum decomposition, which is technically a [math]*[/math]-coalgebra isomorphism, follows from (2). As for the second assertion, this follows from the fact that [math](id\otimes\int_G)v[/math] is the orthogonal projection [math]P_v[/math] onto the space [math]Fix(v)[/math], for any corepresentation [math]v[/math].

(4) Let us define indeed the character of [math]v\in M_n(A)[/math] to be the matrix trace, [math]\chi_v=Tr(v)[/math]. Since this character is a coefficient of [math]v[/math], the orthogonality assertion follows from (3). As for the norm 1 claim, this follows once again from [math](id\otimes\int_G)v=P_v[/math].

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We can now solve a problem that we left open before, namely:

Proposition

The cocommutative Woronowicz algebras appear as the quotients

[[math]] C^*(\Gamma)\to A\to C^*_{red}(\Gamma) [[/math]]

given by [math]A=C^*_\pi(\Gamma)[/math] with [math]\pi\otimes\pi\subset\pi[/math], with [math]\Gamma[/math] being a discrete group.

Show Proof

This follows from the Peter-Weyl theory, and clarifies a number of things said before, notably in Proposition 7.26. Indeed, for a cocommutative Woronowicz algebra the irreducible corepresentations are all 1-dimensional, and this gives the results.

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As another consequence of the above results, once again by following Woronowicz , we have the following statement, dealing with functional analysis aspects, and extending what we already knew about the [math]C^*[/math]-algebras of the usual discrete groups:

Theorem

Let [math]A_{full}[/math] be the enveloping [math]C^*[/math]-algebra of [math]\mathcal A[/math], and [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]. See .

[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 functional analysis. Once again, we refer here to .

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Let us discuss now some interesting examples. Following Wang , we have:

Proposition

The following universal algebras are Woronowicz algebras,

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

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

so the underlying 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, 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]]

Thus, we can indeed define morphisms [math]\Delta,\varepsilon,S[/math] as in Definition 7.25, by using the universal properties of [math]C(O_N^+)[/math], [math]C(U_N^+)[/math], and this gives the result.

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There is a connection here with group duals, coming 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.

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With this notion in hand, we have the following result:

Theorem

The diagonal tori of the basic rotation groups are as follows,

[[math]] \xymatrix@R=15mm@C=15mm{ U_N\ar[r]&U_N^+\\ O_N\ar[r]\ar[u]&O_N^+\ar[u] }\qquad \item[a]ymatrix@R=7mm@C=15mm{\\ :\\} \qquad \item[a]ymatrix@R=14mm@C=15mm{ \mathbb T^N\ar[r]&\widehat{F_N}\\ \mathbb Z_2^N\ar[r]\ar[u]&\widehat{\mathbb Z_2^{*N}}\ar[u] } [[/math]]

where [math]F_N[/math] is the free group on [math]N[/math] generators, and [math]*[/math] is a group-theoretical free product.

Show Proof

This is clear indeed from [math]U_N^+[/math], and the other results can be obtained by imposing to the generators of [math]F_N[/math] the relations defining the corresponding quantum groups.

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As a conclusion to all this, the [math]C^*[/math]-algebra theory suggests developing a theory of “noncommutative geometry”, covering both the classical and the free geometry, by using compact quantum groups. We will be back to this in chapter 8.

General references

Banica, Teo (2024). "Principles of operator algebras". arXiv:2208.03600 [math.OA].