13b. Quantum groups

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

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We will be interested in what follows in the case where the compact quantum space [math]X[/math] is a “compact quantum group”. The axioms for the corresponding [math]C^*[/math]-algebras, found by Woronowicz in , are, in a soft form, 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 [math]\Delta:A\to A\otimes A[/math], [math]\varepsilon:A\to\mathbb C[/math], [math]S:A\to A^{opp}[/math].

The morphisms [math]\Delta,\varepsilon,S[/math] are called comultiplication, counit and antipode. 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]] \begin{eqnarray*} \Delta(\varphi)&=&(g,h)\to \varphi(gh)\\ \varepsilon(\varphi)&=&\varphi(1)\\ S(\varphi)&=&g\to\varphi(g^{-1}) \end{eqnarray*} [[/math]]
• [math]C^*(\Gamma)[/math], with [math]F_N\to\Gamma[/math] finitely generated group. Here the structural maps are:
  [[math]] \begin{eqnarray*} \Delta(g)&=&g\otimes g\\ \varepsilon(g)&=&1\\ S(g)&=&g^{-1} \end{eqnarray*} [[/math]]

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

Show Proof

This is something very standard, the idea being as follows:

(1) Given [math]G\subset U_N[/math], we can set [math]A=C(G)[/math], which is a Woronowicz algebra, together with the matrix [math]u=(u_{ij})[/math] formed by 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]]

Conversely, if [math](A,u)[/math] is a commutative Woronowicz algebra, by using the Gelfand theorem we can write [math]A=C(X)[/math], with [math]X[/math] being a certain compact space. The coordinates [math]u_{ij}[/math] give then an embedding [math]X\subset M_N(\mathbb C)[/math], and since the matrix [math]u=(u_{ij})[/math] is unitary we actually obtain an embedding [math]X\subset U_N[/math], and finally by using the maps [math]\Delta,\varepsilon,S[/math] we conclude that our compact subspace [math]X\subset U_N[/math] is in fact a compact Lie group, as desired.

(2) Consider a finitely generated group [math]F_N\to\Gamma[/math]. We can set [math]A=C^*(\Gamma)[/math], which is by definition the completion of the complex group algebra [math]\mathbb C[\Gamma][/math], with involution given by [math]g^*=g^{-1}[/math], for any [math]g\in\Gamma[/math], with respect to the biggest [math]C^*[/math]-norm, and we obtain a Woronowicz algebra, together with the diagonal matrix formed by the generators of [math]\Gamma[/math]:

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

Conversely, if [math](A,u)[/math] is a cocommutative Woronowicz algebra, the Peter-Weyl theory of Woronowicz, to be explained below, shows that the irreducible corepresentations of [math]A[/math] are all 1-dimensional, and form a group [math]\Gamma[/math], and so we have [math]A=C^*(\Gamma)[/math], as desired.

■

In relation with the above, we should mention that there are actually some analytic subtleties here, coming from amenability, and so our quantum spaces and groups must be divided by a certain equivalence relation, for everything to work fine. To be more precise, in the context of Definition 13.6, we write [math](A,u)=(B,v)[/math] when there is a [math]*[/math]-algebra isomorphism as follows, mapping standard coordinates to standard coordinates:

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

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]] \begin{eqnarray*} (\Delta\otimes id)\Delta&=&(id\otimes \Delta)\Delta\\ (\varepsilon\otimes id)\Delta&=&(id\otimes\varepsilon)\Delta=id \end{eqnarray*} [[/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

The two comultiplication axioms follow from:

[[math]] \begin{eqnarray*} (\Delta\otimes id)\Delta(u_{ij})&=&(id\otimes \Delta)\Delta(u_{ij})=\sum_{kl}u_{ik}\otimes u_{kl}\otimes u_{lj}\\ (\varepsilon\otimes id)\Delta(u_{ij})&=&(id\otimes\varepsilon)\Delta(u_{ij})=u_{ij} \end{eqnarray*} [[/math]]

As for the antipode formulae, the verification here is similar.

■

Summarizing, the Woronowicz algebras appear to have nice properties. In view of Proposition 13.7 and Proposition 13.8, 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},\Gamma=\widehat{G}[/math]. In general, we still agree to write, but in a formal sense:

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

With this in mind, let us call now corepresentation of [math]A[/math] any unitary matrix [math]v\in M_n(A)[/math] satisfying the same conditions as those satisfied by [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 corepresentations can be thought of as corresponding 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 .

■

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]]

We have then the following result, also due to 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].

■

Observe 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 Proposition 13.7.

General references

Banica, Teo (2024). "Invitation to Hadamard matrices". arXiv:1910.06911 [math.CO].