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In the remainder of this chapter we develop the Peter-Weyl theory for the representations of the compact quantum groups, following the paper of Woronowicz . There is quite some work to be done here, and we will do it in two parts, first with some basic algebraic results, which are quite elementary, and then with more advanced results, mixing algebra and analysis. Let us start with the following definition:

Definition

Given two corepresentations [math]v\in M_n(A),w\in M_m(A)[/math], we set

[[math]] Hom(v,w)=\left\{T\in M_{m\times n}(\mathbb C)\Big|Tv=wT\right\} [[/math]]

and we use the following conventions:

• We use the notations [math]Fix(v)=Hom(1,v)[/math], and [math]End(v)=Hom(v,v)[/math].
• We write [math]v\sim w[/math] when [math]Hom(v,w)[/math] contains an invertible element.
• We say that [math]v[/math] is irreducible, and write [math]v\in Irr(G)[/math], when [math]End(v)=\mathbb C1[/math].

In the classical case [math]A=C(G)[/math] we obtain the usual notions concerning the representations. Observe also that in the group dual case we have:

[[math]] g\sim h\iff g=h [[/math]]

Finally, observe that [math]v\sim w[/math] means that [math]v,w[/math] are conjugated by an invertible matrix. Here are a few basic results, regarding the above Hom spaces:

Proposition

We have the following results:

• [math]T\in Hom(u,v),S\in Hom(v,w)\implies ST\in Hom(u,w)[/math].
• [math]S\in Hom(p,q),T\in Hom(v,w)\implies S\otimes T\in Hom(p\otimes v,q\otimes w)[/math].
• [math]T\in Hom(v,w)\implies T^*\in Hom(w,v)[/math].

In other words, the Hom spaces form a tensor [math]*[/math]-category.

Show Proof

These assertions are all elementary, as follows:

(1) By using our assumptions [math]Tu=vT[/math] and [math]Sv=Ws[/math] we obtain, as desired:

[[math]] STu =SvT =wST [[/math]]

(2) Assume indeed that we have [math]Sp=qS[/math] and [math]Tv=wT[/math]. With tensor product notations, as in the proof of Proposition 3.4 above, we have:

[[math]] \begin{eqnarray*} (S\otimes T)(p\otimes v) &=&S_1T_2p_{13}v_{23}\\ &=&(Sp)_{13}(Tv)_{23} \end{eqnarray*} [[/math]]

On the other hand, we have as well the following computation:

[[math]] \begin{eqnarray*} (q\otimes w)(S\otimes T) &=&q_{13}w_{23}S_1T_2\\ &=&(qS)_{13}(wT)_{23} \end{eqnarray*} [[/math]]

The quantities on the right being equal, this gives the result.

(3) By conjugating, and then using the unitarity of [math]v,w[/math], we obtain, as desired:

[[math]] \begin{eqnarray*} Tv=wT &\implies&v^*T^*=T^*w^*\\ &\implies&vv^*T^*w=vT^*w^*w\\ &\implies&T^*w=vT^* \end{eqnarray*} [[/math]]

Finally, the last assertion follows from definitions, and from the obvious fact that, in addition to (1,2,3) above, the Hom spaces are linear spaces, and contain the units. In short, this is just a theoretical remark, that will be used only later on.

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As a main consequence, the spaces [math]End(v)\subset M_n(\mathbb C)[/math] are subalgebras which are stable under [math]*[/math], and so are [math]C^*[/math]-algebras. In order to exploit this fact, we will need a basic result, complementing the operator algebra theory presented in chapter 1 above, namely:

Theorem

Let [math]B\subset M_n(\mathbb C)[/math] be a [math]C^*[/math]-algebra.

• We can write [math]1=p_1+\ldots+p_k[/math], with [math]p_i\in B[/math] central minimal projections.
• Each of the linear spaces [math]B_i=p_iBp_i[/math] is a non-unital [math]*[/math]-subalgebra of [math]B[/math].
• We have a non-unital [math]*[/math]-algebra sum decomposition [math]B=B_1\oplus\ldots\oplus B_k[/math].
• We have unital [math]*[/math]-algebra isomorphisms [math]B_i\simeq M_{r_i}(\mathbb C)[/math], where [math]r_i=rank(p_i)[/math].
• Thus, we have a [math]C^*[/math]-algebra isomorphism [math]B\simeq M_{r_1}(\mathbb C)\oplus\ldots\oplus M_{r_k}(\mathbb C)[/math].

In addition, the final conclusion holds for any finite dimensional [math]C^*[/math]-algebra.

Show Proof

This is something well-known, with the proof of the various assertions in the statement being something elementary, and routine:

(1) This is more of a definition.

(2) This is elementary, coming from [math]p_i^2=p_i=p_i^*[/math].

(3) The verification of the direct sum conditions is indeed elementary.

(4) This follows from the fact that each [math]p_i[/math] was assumed to be central and minimal.

(5) This follows by putting everything together.

As for the last assertion, this follows from (5) by using the GNS representation theorem, which provides us with an embedding [math]B\subset M_n(\mathbb C)[/math], for some [math]n\in\mathbb N[/math].

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Following Woronowicz's paper , we can now formulate a first Peter-Weyl theorem, and to be more precise a first such theorem from a 4-series, as follows:

Theorem (PW1)

Let [math]v\in M_n(A)[/math] be a corepresentation, consider the [math]C^*[/math]-algebra [math]B=End(v)[/math], and write its unit as [math]1=p_1+\ldots+p_k[/math], as above. We have then

[[math]] v=v_1+\ldots+v_k [[/math]]

with each [math]v_i[/math] being an irreducible corepresentation, obtained by restricting [math]v[/math] to [math]Im(p_i)[/math].

Show Proof

This is something very classical, well-known to hold for the compact groups, and the proof in general can be deduced from Theorem 3.8, as follows:

(1) We first associate to our corepresentation [math]v\in M_n(A)[/math] the corresponding coaction map [math]\Phi:\mathbb C^n\to\mathbb C^n\otimes A[/math], given by the following formula:

[[math]] \Phi(e_i)=\sum_je_j\otimes v_{ji} [[/math]]

We say that a linear subspace [math]V\subset\mathbb C^n[/math] is invariant under [math]v[/math] if:

[[math]] \Phi(V)\subset V\otimes A [[/math]]

In this case, we can consider the following restriction map:

[[math]] \Phi_{|V}:V\to V\otimes A [[/math]]

This is a coaction map too, which must come from a subcorepresentation [math]w\subset v[/math].

(2) Consider now a projection [math]p\in End(v)[/math]. From [math]pv=vp[/math] we obtain that the linear space [math]V=Im(p)[/math] is invariant under [math]v[/math], and so this space must come from a subcorepresentation [math]w\subset v[/math]. It is routine to check that the operation [math]p\to w[/math] maps subprojections to subcorepresentations, and minimal projections to irreducible corepresentations.

(3) With these preliminaries in hand, let us decompose the algebra [math]End(v)[/math] as in Theorem 3.8, by using the decomposition of 1 into minimal projections there:

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

Consider now the following vector spaces, obtained as images of these projections:

[[math]] V_i=Im(p_i) [[/math]]

If we denote by [math]v_i\subset v[/math] the subcorepresentations coming from these vector spaces, then we obtain in this way a decomposition [math]v=v_1+\ldots+v_k[/math], as in the statement.

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In order to formulate our second Peter-Weyl type theorem, we will need:

Definition

We denote by [math]u^{\otimes k}[/math], with [math]k=\circ\bullet\bullet\circ\ldots[/math] being a colored integer, the various tensor products between [math]u,\bar{u}[/math], indexed according to the rules

[[math]] u^{\otimes\emptyset}=1\quad,\quad u^{\otimes\circ}=u\quad,\quad u^{\otimes\bullet}=\bar{u} [[/math]]

and multiplicativity, [math]u^{\otimes kl}=u^{\otimes k}\otimes u^{\otimes l}[/math], and call them Peter-Weyl corepresentations.

Here are a few examples of such corepresentations, namely those coming from the colored integers of length 2, to be often used in what follows:

[[math]] u^{\otimes\circ\circ}=u\otimes u\quad,\quad u^{\otimes\circ\bullet}=u\otimes\bar{u} [[/math]]

[[math]] u^{\otimes\bullet\circ}=\bar{u}\otimes u\quad,\quad u^{\otimes\bullet\bullet}=\bar{u}\otimes\bar{u} [[/math]]

There are several particular cases of interest of the above construction, where some considerable simplifications appear, as follows:

Proposition

The Peter-Weyl corepresentations [math]u^{\otimes k}[/math] are as follows:

• In the real case, [math]u=\bar{u}[/math], we can assume [math]k\in\mathbb N[/math].
• In the classical case, we can assume, up to equivalence, [math]k\in\mathbb N\times\mathbb N[/math].

Show Proof

These assertions are both elementary, as follows:

(1) Here we have indeed [math]u^{\otimes k}=u^{\otimes|k|}[/math], where [math]|k|\in\mathbb N[/math] is the length. Thus the Peter-Weyl corepresentations are indexed by [math]\mathbb N[/math], as claimed.

(2) In the classical case, our claim is that we have equivalences [math]v\otimes w\sim w\otimes v[/math], implemented by the flip operator [math]\Sigma(a\otimes b)=b\otimes a[/math]. Indeed, we have:

[[math]] \begin{eqnarray*} v\otimes w &=&v_{13}w_{23}\\ &=&w_{23}v_{13}\\ &=&\Sigma w_{13}v_{23}\Sigma\\ &=&\Sigma(w\otimes v)\Sigma \end{eqnarray*} [[/math]]

In particular we have an equivalence [math]u\otimes\bar{u}\sim\bar{u}\otimes u[/math]. We conclude that the Peter-Weyl corepresentations are the corepresentations of type [math]u^{\otimes k}\otimes\bar{u}^{\otimes l}[/math], with [math]k,l\in\mathbb N[/math].

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Here is now our second Peter-Weyl theorem, from a series of a total 4 Peter-Weyl theorems, also from Woronowicz , complementing Theorem 3.9 above:

Theorem (PW2)

Each irreducible corepresentation of [math]A[/math] appears as:

[[math]] v\subset u^{\otimes k} [[/math]]

That is, [math]v[/math] appears inside a certain Peter-Weyl corepresentation.

Show Proof

Given an arbitrary corepresentation [math]v\in M_n(A)[/math], consider its space of coefficients, [math]C(v)=span(v_{ij})[/math]. It is routine to check that the construction [math]v\to C(v)[/math] is functorial, in the sense that it maps subcorepresentations into subspaces. By definition of the Peter-Weyl corepresentations, we have:

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

Now given a 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 [math]k_1,\ldots,k_p[/math]:

[[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]]

Together with Theorem 3.9, this leads to the conclusion in the statement.

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General references

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