3a. Representations

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

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In order to reach to some more advanced insight into the structure of the compact quantum groups, we can use representation theory. We follow Woronowicz's paper , with a few simplifications coming from our [math]S^2=id[/math] formalism. We first have:

Definition

A corepresentation of a Woronowicz algebra [math](A,u)[/math] is a unitary matrix [math]v\in M_n(\mathcal A)[/math] over the dense [math]*[/math]-algebra [math]\mathcal A= \lt u_{ij} \gt [/math], satisfying:

[[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]]

That is, [math]v[/math] must satisfy the same conditions as [math]u[/math].

As basic examples here, we have the trivial corepresentation, having dimension 1, as well as the fundamental corepresentation, and its adjoint:

[[math]] 1=(1)\quad,\quad u=(u_{ij})\quad,\quad \bar{u}=(u_{ij}^*) [[/math]]

In the classical case, we recover in this way the usual representations of [math]G[/math]:

Proposition

Given a closed subgroup [math]G\subset U_N[/math], the corepresentations of the associated Woronowicz algebra [math]C(G)[/math] are in one-to-one correspondence, given by

[[math]] \pi(g)=\begin{pmatrix}v_{11}(g)&\ldots&v_{1n}(g)\\ \vdots&&\vdots\\ v_{n1}(g)&\ldots&v_{nn}(g) \end{pmatrix} [[/math]]

with the finite dimensional unitary smooth representations of [math]G[/math].

Show Proof

With [math]A=C(G)[/math], consider the unitary matrices [math]v\in M_n(A)[/math] satisfying the equations in Definition 3.1. By using the computations from chapter 2, performed when proving that any closed subgroup [math]G\subset U_N[/math] is indeed a compact quantum group, we conclude that we have a correspondence [math]v\leftrightarrow\pi[/math] as in the statement, between such matrices, and the finite dimensional unitary representations of [math]G[/math].

Regarding now the smoothness part, this is something more subtle, which requires some knowledge of Lie theory. The point is that any closed subgroup [math]G\subset U_N[/math] is a Lie group, and since the coefficient functions [math]u_{ij}:G\to\mathbb C[/math] are smooth, we have:

[[math]] \mathcal A\subset C^\infty(G) [[/math]]

Thus, when assuming [math]v\in M_n(\mathcal A)[/math], the corresponding representation [math]\pi:G\to U_n[/math] is smooth, and the converse of this fact is known to hold as well.

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In general now, we have the following operations on the corepresentations:

Proposition

The corepresentations are subject to the following operations:

• Making sums, [math]v+w=diag(v,w)[/math].
• Making tensor products, [math](v\otimes w)_{ia,jb}=v_{ij}w_{ab}[/math].
• Taking conjugates, [math](\bar{v})_{ij}=v_{ij}^*[/math].

Show Proof

Observe that the result holds in the commutative case, where we obtain the usual operations on the representations of the corresponding group. In general now:

(1) Everything here is clear, as already mentioned in chapter 2 above, when using such corepresentations in order to construct quantum group quotients.

(2) First of all, the matrix [math]v\otimes w[/math] is unitary. Indeed, we have:

[[math]] \begin{eqnarray*} \sum_{jb}(v\otimes w)_{ia,jb}(v\otimes w)_{kc,jb}^* &=&\sum_{jb}v_{ij}w_{ab}w_{cb}^*v_{kj}^*\\ &=&\delta_{ac}\sum_jv_{ij}v_{kj}^*\\ &=&\delta_{ik}\delta_{ac} \end{eqnarray*} [[/math]]

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

[[math]] \begin{eqnarray*} \sum_{jb}(v\otimes w)^*_{jb,ia}(v\otimes w)_{jb,kc} &=&\sum_{jb}w_{ba}^*v_{ji}^*v_{jk}w_{bc}\\ &=&\delta_{ik}\sum_bw_{ba}^*w_{bc}\\ &=&\delta_{ik}\delta_{ac} \end{eqnarray*} [[/math]]

The comultiplicativity condition follows from the following computation:

[[math]] \begin{eqnarray*} \Delta((v\otimes w)_{ia,jb}) &=&\sum_{kc}v_{ik}w_{ac}\otimes v_{kj}w_{cb}\\ &=&\sum_{kc}(v\otimes w)_{ia,kc}\otimes(v\otimes w)_{kc,jb} \end{eqnarray*} [[/math]]

The proof of the counitality condition is similar, as follows:

[[math]] \varepsilon((v\otimes w)_{ia,jb}) =\delta_{ij}\delta_{ab} =\delta_{ia,jb} [[/math]]

As for the condition involving the antipode, this can be checked as follows:

[[math]] S((v\otimes w)_{ia,jb}) =w_{ba}^*v_{ji}^* =(v\otimes w)_{jb,ia}^* [[/math]]

(3) In order to check that [math]\bar{v}[/math] is unitary, we can use the antipode, exactly as we did in chapter 2 above, for [math]\bar{u}[/math]. As for the comultiplicativity axioms, these are all clear.

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We have as well the following supplementary operation:

Proposition

Given a corepresentation [math]v\in M_n(A)[/math], its spinned version

[[math]] w= UvU^* [[/math]]

is a corepresentation as well, for any unitary matrix [math]U\in U_n[/math].

Show Proof

The matrix [math]w[/math] is unitary, and its comultiplicativity properties can be checked by doing some computations. Here is however another proof of this fact, using a useful trick. In the context of Definition 3.1, if we write [math]v\in M_n(\mathbb C)\otimes A[/math], the axioms read:

[[math]] (id\otimes\Delta)v=v_{12}v_{13} [[/math]]

[[math]] (id\otimes\varepsilon)v=1 [[/math]]

[[math]] (id\otimes S)v=v^* [[/math]]

Here we use standard tensor calculus conventions. Now when spinning by a unitary the matrix that we obtain, with these conventions, is [math]w=U_1vU_1^*[/math], and we have:

[[math]] \begin{eqnarray*} (id\otimes\Delta)w &=&U_1v_{12}v_{13}U_1^*\\ &=&U_1v_{12}U_1^*\cdot U_1v_{13}U_1^*\\ &=&w_{12}w_{13} \end{eqnarray*} [[/math]]

The proof of the counitality condition is similar, as follows:

[[math]] (id\otimes\varepsilon)w =U\cdot 1\cdot U =1 [[/math]]

Finally, the last condition, involving the antipode, can be checked as follows:

[[math]] (id\otimes S)w =U_1v^*U_1^* =w^* [[/math]]

Thus, with usual notations, [math]w=UvU^*[/math] is a corepresentation, as claimed.

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As a philosophical comment here, the above proof might suggest that the more abstract our notations and formalism, and our methods in order to deal with mathematical questions, the easier our problems will become. But this is wrong. Bases and indices are a blessing: they can be understood by undergraduate students, computers, fellow scientists, engineers, and of course also by yourself, when you're tired or so.

In addition, in the quantum group context, we will see later on, starting from chapter 4 below, that bases and indices can be turned into something very beautiful and powerful, allowing us to do some serious theory, well beyond the level of abstractions.

Back to work now, in the group dual case, we have the following result:

Proposition

Assume [math]A=C^*(\Gamma)[/math], with [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math] being a discrete group.

• Any group element [math]h\in\Gamma[/math] is a [math]1[/math]-dimensional corepresentation of [math]A[/math], and the operations on corepresentations are the usual ones on group elements.
• Any diagonal matrix of type [math]v=diag(h_1,\ldots,h_n)[/math], with [math]n\in\mathbb N[/math] arbitrary, and with [math]h_1,\ldots,h_n\in\Gamma[/math], is a corepresentation of [math]A[/math].
• More generally, any matrix of type [math]w=Udiag(h_1,\ldots,h_n)U^*[/math] with [math]h_1,\ldots,h_n\in\Gamma[/math] and with [math]U\in U_n[/math], is a corepresentation of [math]A[/math].

Show Proof

These assertions are all elementary, as follows:

(1) The first assertion is clear from definitions and from the comultiplication, counit and antipode formulae for the discrete group algebras, namely:

[[math]] \Delta(h)=h\otimes h [[/math]]

[[math]] \varepsilon(h)=1 [[/math]]

[[math]] S(h)=h^{-1} [[/math]]

The assertion on the operations is clear too, because we have:

[[math]] (g)\otimes(h)=(gh) [[/math]]

[[math]] \overline{(g)}=(g^{-1}) [[/math]]

(2) This follows from (1) by performing sums, as in Proposition 3.3 above.

(3) This follows from (2) and from the fact that we can conjugate any corepresentation by a unitary matrix, as explained in Proposition 3.4 above.

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Observe that the class of corepresentations in (3) is stable under all the operations from Propositions 3.3, and under the spinning operation from Proposition 3.4 too. When [math]\Gamma[/math] is abelian we can apply Proposition 3.2 with [math]G=\widehat{\Gamma}[/math], and after performing a number of identifications, we conclude that these are all the corepresentations of [math]C^*(\Gamma)[/math].

We will see later that this latter fact holds in fact for any discrete group [math]\Gamma[/math]. To be more precise, this is something non-trivial, which will follow from Peter-Weyl theory.

Summarizing, the representations of a compact quantum group can be defined as in the classical case, but by using coefficients, and in the group dual case we obtain something which is a priori quite simple too, namely formal direct sums of group elements.

General references

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