3d. More Peter-Weyl

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Let us go back now to algebra, and establish two more Peter-Weyl theorems. We will need the following result, which is very useful, and is of independent interest:

Theorem

We have a Frobenius type isomorphism

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

valid for any two corepresentations [math]v,w[/math].

Show Proof

According to the definitions, we have the following equivalence:

[[math]] \begin{eqnarray*} T\in Hom(v,w) &\iff&Tv=wT\\ &\iff&\sum_jT_{aj}v_{ji}=\sum_bw_{ab}T_{bi} \end{eqnarray*} [[/math]]

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

[[math]] \begin{eqnarray*} T\in Fix(\bar{v}\otimes w) &\iff&(\bar{v}\otimes w)T=T\\ &\iff&\sum_{kb}v_{ik}^*w_{ab}T_{bk}=T_{ai} \end{eqnarray*} [[/math]]

With these formulae in hand, we must prove that we have:

[[math]] \sum_jT_{aj}v_{ji}=\sum_bw_{ab}T_{bi}\iff \sum_{kb}v_{ik}^*w_{ab}T_{bk}=T_{ai} [[/math]]

(1) In one sense, the computation is as follows, using the unitarity of [math]v^t[/math]:

[[math]] \begin{eqnarray*} \sum_{kb}v_{ik}^*w_{ab}T_{bk} &=&\sum_kv_{ik}^*\sum_bw_{ab}T_{bk}\\ &=&\sum_kv_{ik}^*\sum_jT_{aj}v_{jk}\\ &=&\sum_j(\bar{v}v^t)_{ij}T_{aj}\\ &=&T_{ai} \end{eqnarray*} [[/math]]

(2) In the other sense we have, once again by using the unitarity of [math]v^t[/math]:

[[math]] \begin{eqnarray*} \sum_jT_{aj}v_{ji} &=&\sum_jv_{ji}\sum_{kb}v_{jk}^*w_{ab}T_{bk}\\ &=&\sum_{kb}(v^t\bar{v})_{ik}w_{ab}T_{bk}\\ &=&\sum_bw_{ab}T_{bi} \end{eqnarray*} [[/math]]

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

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With these ingredients, namely two Peter-Weyl theorems, Haar measure and Frobenius duality, we can establish a third Peter-Weyl theorem, also from Woronowicz :

Theorem (PW3)

The dense subalgebra [math]\mathcal A\subset A[/math] decomposes as a direct sum

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

with this being an isomorphism of [math]*[/math]-coalgebras, and with the summands being pairwise orthogonal with respect to the scalar product given by

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

where [math]\int_G[/math] is the Haar integration over [math]G[/math].

Show Proof

By combining the previous Peter-Weyl results, from Theorem 3.9 and Theorem 3.12 above, we deduce that we have 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]]

Thus, 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]]

But this follows from Theorem 3.18, via Theorem 3.21. Let us set indeed:

[[math]] P_{ia,jb}=\int_Gv_{ij}w_{ab}^* [[/math]]

Then [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 this gives the result.

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We can obtain further results by using characters, which are defined as follows:

Proposition

The characters of the corepresentations, given by

[[math]] \chi_v=\sum_iv_{ii} [[/math]]

behave as follows, in respect to the various operations:

[[math]] \chi_{v+w}=\chi_v+\chi_w [[/math]]

[[math]] \chi_{v\otimes w}=\chi_v\chi_w [[/math]]

[[math]] \chi_{\bar{v}}=\chi_v^* [[/math]]

In addition, given two equivalent corepresentations, [math]v\sim w[/math], we have [math]\chi_v=\chi_w[/math].

Show Proof

The three formulae in the statement are all clear from definitions. Regarding now the last assertion, assuming that we have [math]v=T^{-1}wT[/math], we obtain:

[[math]] \begin{eqnarray*} \chi_v &=&Tr(v)\\ &=&Tr(T^{-1}wT)\\ &=&Tr(w)\\ &=&\chi_w \end{eqnarray*} [[/math]]

We conclude that [math]v\sim w[/math] implies [math]\chi_v=\chi_w[/math], as claimed.

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We have the following more advanced result, regarding the characters, also from Woronowicz , completing the Peter-Weyl theory:

Theorem (PW4)

The characters of the irreducible corepresentations belong to the [math]*[/math]-algebra

[[math]] \mathcal A_{central}=\left\{a\in\mathcal A\Big|\Sigma\Delta(a)=\Delta(a)\right\} [[/math]]

of “smooth central functions” on [math]G[/math], and form an orthonormal basis of it.

Show Proof

As a first remark, the linear space [math]\mathcal A_{central}[/math] defined above is indeed an algebra. In the classical case, we obtain in this way the usual algebra of smooth central functions. Also, in the group dual case, where we have [math]\Sigma\Delta=\Delta[/math], we obtain the whole convolution algebra. Regarding now the proof, in general, this goes as follows:

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

[[math]] \begin{eqnarray*} \Sigma\Delta(\chi_v) &=&\Sigma\left(\sum_{ij}v_{ij}\otimes v_{ji}\right)\\ &=&\sum_{ij}v_{ji}\otimes v_{ij}\\ &=&\Delta(\chi_v) \end{eqnarray*} [[/math]]

(2) Conversely, consider an element [math]a\in\mathcal A[/math], written as follows:

[[math]] a=\sum_{v\in Irr(A)}a_v [[/math]]

The condition [math]a\in\mathcal A_{central}[/math] is then equivalent to the following conditions:

[[math]] a_v\in\mathcal A_{central}\quad,\forall v\in Irr(A) [[/math]]

But each condition [math]a_v\in\mathcal A_{central}[/math] means that [math]a_v[/math] must be a scalar multiple of the corresponding character [math]\chi_v[/math], and so the characters form a basis of [math]\mathcal A_{central}[/math], as stated.

(3) The fact that we have an orthogonal basis follows from Theorem 3.22.

(4) Finally, regarding the norm 1 assertion, consider the following integrals:

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

We know from Theorem 3.18 that these integrals form the orthogonal projection onto the following vector space, computed via Theorem 3.21:

[[math]] Fix(v\otimes\bar{v}) \simeq End(\bar{v}) =\mathbb C1 [[/math]]

By using this fact, we obtain the following formula:

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

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

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As a first application of the Peter-Weyl theory, and more specifically of Theorem 3.24, we can now clarify a question that we left open in chapter 2, regarding the cocommutative case. To be more precise, once again following Woronowicz , we have:

Theorem

For a Woronowicz algebra [math]A[/math], the following are equivalent:

• [math]A[/math] is cocommutative, [math]\Sigma\Delta=\Delta[/math].
• The irreducible corepresentations of [math]A[/math] are all [math]1[/math]-dimensional.
• [math]A=C^*(\Gamma)[/math], for some group [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math], up to equivalence.

Show Proof

This follows from the Peter-Weyl theory, as follows:

[math](1)\implies(2)[/math] The assumption [math]\Sigma\Delta=\Delta[/math] tells us that the inclusion [math]\mathcal A_{central}\subset\mathcal A[/math] is an isomorphism, and by using Theorem 3.24 we conclude that any irreducible corepresentation of [math]A[/math] must be equal to its character, and so must be 1-dimensional.

[math](2)\implies(3)[/math] This follows once again from Peter-Weyl, because if we denote by [math]\Gamma[/math] the group formed by the 1-dimensional corepresentations, then we have [math]\mathcal A=\mathbb C[\Gamma][/math], and so [math]A=C^*(\Gamma)[/math] up to the standard equivalence relation for Woronowicz algebras.

[math](3)\implies(1)[/math] This is something trivial, that we already know from chapter 2.

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The above result is not the end of the story, because one can still ask what happens, without reference to the equivalence relation. We will be back to this later.

At the level of the product operations, we have, following Wang :

Proposition

We have the following results:

• The irreducible corepresentations of [math]C(G\times H)[/math] are the tensor products of the form [math]v\otimes w[/math], with [math]v,w[/math] being irreducible corepresentations of [math]C(G),C(H)[/math].
• The irreducible corepresentations of [math]C(G\,\hat{*}\,H)[/math] appear as alternating tensor products of irreducible corepresentations of [math]C(G)[/math] and of [math]C(H)[/math].
• The irreducible corepresentations of [math]C(H)\subset C(G)[/math] are the irreducible corepresentations of [math]C(G)[/math] whose coefficients belong to [math]C(H)[/math].
• The irreducible corepresentations of [math]C(PG)\subset C(G)[/math] are the irreducible corepresentations of [math]C(G)[/math] which appear by decomposing the tensor powers of [math]u\otimes\bar{u}[/math].

Show Proof

This is something routine, the idea being as follows:

(1) Here we can integrate characters, by using Proposition 3.19 (1), and we conclude that if [math]v,w[/math] are irreducible corepresentations of [math]C(G),C(H)[/math], then [math]v\otimes w[/math] is an irreducible corepresentation of [math]C(G\times H)[/math]. Now since the coefficients of these latter corepresentations span [math]\mathcal C(G\times H)[/math], by Peter-Weyl these are all the irreducible corepresentations.

(2) Here we can use a similar method. By using Proposition 3.19 (2) we conclude that if [math]v_1,v_2,\ldots[/math] are irreducible corepresentations of [math]C(G)[/math] and [math]w_1,w_2,\ldots[/math] are irreducible corepresentations of [math]C(H)[/math], then [math]v_1\otimes w_1\otimes v_2\otimes w_2\otimes\ldots[/math] is an irreducible corepresentation of [math]C(G\,\hat{*}\,H)[/math], and then we can conclude by using the Peter-Weyl theory.

(3) This is clear from definitions, and from the Peter-Weyl theory.

(4) This is a particular case of the result (3) above.

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Let us go back now to Theorem 3.25, and try to understand what happens in general, without reference to the equivalence relation. We know from chapter 1 that associated to any discrete group [math]\Gamma[/math] are at least two group algebras, which are as follows:

[[math]] C^*(\Gamma)\to C^*_{red}(\Gamma)\subset B(l^2(\Gamma)) [[/math]]

For the finite, or abelian, or more generally amenable groups [math]\Gamma[/math], these two algebras are known to coincide, but in the non-amenable case, the opposite happens. Thus, we are led into the question on whether [math]C^*_{red}(\Gamma)[/math], and other possible group algebras of [math]\Gamma[/math], are Woronowicz algebras in our sense, having morphisms as follows:

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

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

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

Generally speaking, the answer here is “no”, and the subject is quite technical, requiring a good knowledge of advanced functional analysis. In order to have [math]C^*_{red}(\Gamma)[/math] among our examples, if we really want to, we must change a bit our axioms, as follows:

Proposition

Given a discrete group [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math], its reduced algebra [math]A=C^*_{red}(\Gamma)[/math] has morphisms as follows, given on generators by the usual formulae,

[[math]] \Delta:A\to A\otimes_{min}A [[/math]]

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

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

where [math]\otimes_{min}[/math] is the spatial tensor product of [math]C^*[/math]-algebras, and where [math]\mathcal A=\mathbb C[\Gamma][/math].

Show Proof

This is something quite technical, and philosophical, related to [math]*[/math]-algebras vs [math]C^*[/math]-algebras, and to [math]\otimes_{min}[/math] vs [math]\otimes_{max}[/math], that we will not really need in what follows. In what regards the comultiplication, consider the following diagonal embedding:

[[math]] \Gamma\subset\Gamma\times\Gamma\quad,\quad g\to(g,g) [[/math]]

This embedding induces a [math]*[/math]-algebra representation, as follows:

[[math]] \mathbb C[\Gamma]\to B(l^2(\Gamma))\otimes_{min}B(l^2(\Gamma))\quad,\quad g\to g\otimes g [[/math]]

We can extend then this representation into a morphism [math]\Delta[/math], as in the statement. As for the existence of morphisms [math]\varepsilon,S[/math] as in the statement, this is clear.

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Going ahead with some philosophy, the above result might suggest to modify our quantum group axioms, in a somewhat obvious way, with a densely defined counit, as to include algebras of type [math]C^*_{red}(\Gamma)[/math] in our formalism. But do we really want to do that. Remember, we are interested here in quantum spaces and quantum groups, which are well-defined up to equivalence, and so Theorem 3.25 above is all we need.

What does make sense, however, is to do such modifications in case you are interested in more general quantum groups, such as the Drinfeld-Jimbo deformations at [math]q \gt 0[/math], which are not covered by our formalism, and this is what Woronowicz did in . But, as explained in chapter 2, from a modern perspective at least, these deformations with parameter [math]q \gt 0[/math] have only theoretical interest, and we will surely not follow this way.

Let us discuss now, however, in relation with all this, the notion of amenability, which is something important and useful. The basic result here, due to Blanchard , once again requiring a good knowledge of functional analysis, is as follows:

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\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)\implies(2)[/math] This 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)\implies(3)[/math] This is trivial, because we have quotient maps [math]A_{full}\to A\to A_{red}[/math], and so our assumption [math]A_{full}=A_{red}[/math] implies that we have [math]A=A_{red}[/math].

[math](3)\implies(4)[/math] This implication is clear too, because we have:

[[math]] \begin{eqnarray*} \varepsilon(Re(\chi_u)) &=&\frac{1}{2}\left(\sum_{i=1}^N\varepsilon(u_{ii})+\sum_{i=1}^N\varepsilon(u_{ii}^*)\right)\\ &=&\frac{1}{2}(N+N)\\ &=&N \end{eqnarray*} [[/math]]

Thus the element [math]N-Re(\chi_u)[/math] is not invertible in [math]A_{red}[/math], as claimed.

[math](4)\implies(1)[/math] In terms of the corepresentation [math]v=u+\bar{u}[/math], whose dimension is [math]2N[/math] and whose character is [math]2Re(\chi_u)[/math], our assumption [math]N\in\sigma(Re(\chi_u))[/math] reads:

[[math]] \dim v\in\sigma(\chi_v) [[/math]]

By functional calculus the same must hold for [math]w=v+1[/math], and then once again by functional calculus, the same must hold for any tensor power of [math]w[/math]:

[[math]] w_k=w^{\otimes k} [[/math]]

Now choose for each [math]k\in\mathbb N[/math] a state [math]\varepsilon_k\in A_{red}^*[/math] having the following property:

[[math]] \varepsilon_k(w_k)=\dim w_k [[/math]]

By Peter-Weyl we must have [math]\varepsilon_k(r)=\dim r[/math] for any [math]r\leq w_k[/math], and since any irreducible corepresentation appears in this way, the sequence [math]\varepsilon_k[/math] converges to a counit map:

[[math]] \varepsilon:A_{red}\to\mathbb C [[/math]]

In order to finish, we can use the right regular corepresentation. Indeed, we can define such a corepresentation by the following formula:

[[math]] W(a\otimes x)=\Delta(a)(1\otimes x) [[/math]]

This corepresentation is unitary, so we can define a morphism as follows:

[[math]] \Delta':A_{red}\to A_{red}\otimes A_{full} [[/math]]

[[math]] a\to W(a\otimes1)W^* [[/math]]

Now by composing with [math]\varepsilon\otimes id[/math], we obtain a morphism as follows:

[[math]] (\varepsilon\otimes id)\Delta':A_{red}\to A_{full} [[/math]]

[[math]] u_{ij}\to u_{ij} [[/math]]

Thus, we have our inverse map for the projection [math]A_{full}\to A_{red}[/math], as desired.

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All the above was of course quite short, but we will be back to this, with full details, and with a systematic study of the notion of amenability, in chapter 14 below. In particular, we will discuss in detail the case of the usual discrete group algebras [math]A=C^*(\Gamma)[/math], by further building on the findings in Theorem 3.25 and Proposition 3.27.

Here are now some basic applications of the above amenability result:

Proposition

We have the following results:

• The compact Lie groups [math]G\subset U_N[/math] are all coamenable.
• A group dual [math]G=\widehat{\Gamma}[/math] is coamenable precisely when [math]\Gamma[/math] is amenable.
• A product [math]G\times H[/math] of coamenable compact quantum groups is coamenable.

Show Proof

This follows indeed from the results that we have:

(1) This is clear by using any of the criteria in Theorem 3.28 above, because for an algebra of type [math]A=C(G)[/math], we have [math]A_{full}=A_{red}[/math].

(2) Here the various criteria in Theorem 3.28 above correspond to the various equivalent definitions of the amenability of a discrete group.

(3) This follows from the description of the Haar functional of [math]C(G\times H)[/math], from Proposition 3.19 (1) above. Indeed, if [math]\int_G,\int_H[/math] are both faithful, then so is [math]\int_G\otimes\int_H[/math].

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As already mentioned, we will be back to this, in chapter 14 below. But that is in a long time from now, so perhaps time for some philosophy, and advice, in relation with various functional analysis issues, including tensor products, and amenability:

(1) Sorry for having to start with this, but I'm sure that there might be a graduate student or postdoc around you, or perhaps even young researcher, struggling with tensor products and amenability, and telling you something of type “operator algebras are all about tensor products and amenability, you ain't understand anything, you have to spend time and learn tensor products and amenability first, before anything else.

(2) Which is utterly wrong, and believe me, senior researcher talking here. Operator algebras are about quantum mechanics, and more specifically about modern quantum mechanics, from the 1950s onwards. And here, what's needed are quantum groups and quantum spaces, axiomatized as we did it, via equivalence relation between the corresponding operator algebras, and with minimal fuss about tensor products and amenability.

(3) Of course tensor products and amenability will come into play, at some point. But later. That's advanced. More precisely, amenability comes into play, be that in modern mathematics, or modern physics, via Theorem 3.28 (4), called Kesten criterion, and some further work that can be done on that, involving random walks, spectral measures and so on. But that's advanced level, conformal field theory (CFT), or higher.

(4) So stay with me, and of course we'll talk about such things, in due time, meaning end of this book, chapters 13-16 below. And to that graduate student friend of yours, please tell him that he's on the good track of revolutionizing quantum mechanics from the 1920s. And more specifically, from the early 1920s. Which is of course a decent business, papers about old quantum mechanics being always welcome for publication.

Of course, if the advice about learning tensor products and amenability comes from your PhD advisor, guess we'll have to do that. Which is not a problem, that will take you 1 month or so, and you might learn some interesting things there. That you can effectively use later on, once you'll have the black belt in mathematics and physics.

General references

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