Quantum Clifford theory

# Quantum actions on discrete quantum spaces and a generalization of Clifford’s theory of representations

Kenny De Commer Vakgroep Wiskunde, Vrije Universiteit Brussel, Belgium Paweł Kasprzak Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Poland Adam Skalski Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland  and  Piotr M. Sołtan Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Poland
###### Abstract.

To any action of a compact quantum group on a von Neumann algebra which is a direct sum of factors we associate an equivalence relation corresponding to the partition of a space into orbits of the action. We show that in case all factors are finite-dimensional (i.e. when the action is on a discrete quantum space) the relation has finite orbits. We then apply this to generalize the classical theory of Clifford, concerning the restrictions of representations to normal subgroups, to the framework of quantum subgroups of discrete quantum groups, itself extending the context of closed normal quantum subgroups of compact quantum groups. Finally, a link is made between our equivalence relation in question and another equivalence relation defined by R. Vergnioux.

###### Key words and phrases:
compact quantum group; discrete quantum space; Clifford’s theory; irreducible representations
###### 2010 Mathematics Subject Classification:
Primary: 46L89, Secondary: 20G42, 20C99

## 1. Introduction

The story and motivation behind this paper mirror to an extent those behind its classical, almost eighty years old predecessor: Clifford theory, as developed in the article [clifford], was clearly inspired by Frobenius’s induced representations. Let us recall that [clifford] concerns questions around the decomposition of the restriction of a given irreducible representation of a group to a normal subgroup (for a modern treatment of the results of that paper for finite groups we refer to [Ceccherini]). Clifford considers general groups, but only finite-dimensional representations. We will mostly focus on the compact case, where finite dimensionality is automatic. The key notion appearing in this context is that of orbits of the adjoint action of on , the set of (equivalence classes of) irreducible representations of .

Since the arrival of compact quantum groups on the scene in the 1980s and the development of a satisfactory theory of locally compact quantum groups at the turn of the century, there has been a lot of interest in the study of various aspects of representation theory in this, quantum, context. In particular the articles [KustermansInduced] and [Vaes-induction] developed in full a quantum counterpart of the induction theory of Mackey – itself generalizing the aforementioned work of Frobenius for finite groups. The induction mechanisms proposed by Kustermans and Vaes are necessarily somewhat complicated, and in [VergniouxVoigt] a simpler picture was established for discrete quantum groups. On the other hand, in the recent article [KalKS] a notion of an open quantum subgroup was proposed (encompassing in particular all quantum subgroups of discrete quantum groups), allowing for a framework in which one can use Rieffel’s approach to induction ([KKSinduced]). All these developments made natural the need to understand better the relation between open and discrete induction, as studied respectively in [KKSinduced] and [VergniouxVoigt], and brought us to developing a quantum counterpart of Clifford theory.

Classical Clifford theory deals with the relationship between representations of a group and their restrictions to a normal subgroup of , the main tool being the action of on irreducible representations of by composition with a -inner automorphism. In our approach we first look at a pair consisting of a compact quantum group and its closed normal quantum subgroup . This pair gives rise to a discrete quantum group with a quantum subgroup . The quantum subgroup is normal ([extensions, Page 46]) and we can recover as and . Restriction of representations from to translates to restricting representations of to the subalgebra . Our generalization will consist of dropping the assumption of normality of . In particular this introduces a difference between left and right quotients and and it turns out that is more suited to our conventions. This difference, however, is not of importance, and in any case we have , where is the unitary antipode of . In this context we will introduce an action of the compact quantum group on corresponding to the action known from Clifford theory, and introduce the notion of orbit for this action. Then we will be in a position to prove a quantum analog of Clifford’s theorem on restriction of irreducible representations.

Let us briefly describe the contents of the paper. In Section 2 we introduce the necessary language and notation as well as list some standard results from the theory of compact and discrete quantum groups. Section 3 contains a proof of a fundamental result saying that if a compact quantum group acts ergodically on a von Neumann algebra with a finite-dimensional direct summand then itself must be finite-dimensional (Theorem 3.4). In Section 4 we introduce and study the quantum group analog of the relation of being in the same orbit with respect to a compact quantum group action on a direct sum of factors. In Section 5.1 we give two applications of the theory. First we generalize the main result of Clifford theory concerning restrictions of representations to a normal subgroup. Then, in Section 5.2 we exhibit the connection of the equivalence arising in quantum Clifford theory with an equivalence relation of Vergnioux ([orientation]).

## 2. Notation and preliminaries

For a von Neumann algebra the Banach space of normal functionals on will be denoted by , and for we define by the standard formula for all . The center of will be denoted . The tensor product of von Neumann algebras will be denoted while the symbol will be reserved for minimal tensor products of -algebras and for tensor products of maps in various settings. On one occasion we will be considering a tensor product of a von Neumann algebra with a -weakly closed operator space. The symbol will in this case denote the -weak completion of the algebraic tensor product in its natural spatial implementation.

All scalar products will be linear on the right. We will often use Sweedler notation and leg numbering notation familiar from Hopf algebra theory and quantum group literature. For a Hilbert space and vectors the symbol will denote the continuous functional on given by the formula .

Given a family of -algebras the symbol will denote the -direct sum of the family , while will be the -direct sum of the same family. We remark that if is a family of matrix algebras then any (not necessarily unital) -weakly closed -subalgebra of is again isomorphic to a product of matrix algebras.

Throughout the paper will denote a compact quantum group. We will study it mainly via the associated von Neumann algebra , representing the “algebra of bounded measurable functions on ”, equipped with coproduct . The Haar state on will be denoted by and is assumed to act on the GNS space of denoted . A (finite-dimensional) unitary representation of on a (finite-dimensional) Hilbert space is a unitary matrix such that . Its coefficients are elements of the form , with . There are natural notions of unitary equivalence and irreducibility of representations of compact quantum groups ([cqg]), and any irreducible representation is finite-dimensional. We will denote by the set of all equivalence classes of irreducible representations of and tacitly assume that for every class a representative has been chosen and fixed. We will often denote this representative by the same symbol as the class itself. Furthermore, for the symbol denotes the (classical) dimension of the corresponding representation. The keystone of Woronowicz’s quantum Peter-Weyl theory is the fact that coefficients of irreducible representations of span a dense unital -subalgebra of , denoted , and is a Hopf -algebra, whose antipode we will denote by . We say that is finite if (equivalently ) is finite-dimensional, and that is of Kac type if is a tracial state.

The algebra is also equipped with a map , called the unitary antipode, leaving invariant, which may be used to define the contragredient representation of a given representation , see [pseudogr, cqg]. There is also a natural notion of a tensor product of two representations, which we denote . The operations of taking contragredient representations and tensor product pass to equivalence classes.

To every compact quantum group we associate the dual discrete quantum group , which is usually studied either via its algebra of “functions vanishing at infinity” or its algebra of “bounded functions”. Both of these algebras are then equipped with a natural coproduct , allowing to take the tensor product of (normal) algebra representations, and a unitary antipode . There is then a natural one-to-one correspondence between (irreducible) representations of and (irreducible) normal representations of the von Neumann algebra , respecting tensor products and contragredients of representations. More precisely, each representation of is of the form for a normal representation of , where is the right regular representation ([q-lorentz, Section 3]). Then tensor product of and is and up to equivalence, with the transpose operation. This justifies the notation introduced above, and in what follows we will identify with the corresponding representation .

By an action of a compact quantum group on a von Neumann algebra we mean an injective, normal unital -homomorphism satisfying the condition

 (α⊗id\mathnormalL∞(G))∘α=(idM⊗ΔG)∘α. (2.1)

We shall often write for , thus defining a “quantum (measure) space” . We will also write for the algebraic core of (see [KennyLectures, podles]). It is a dense unital -subalgebra of (called also the Podleś subalgebra) such that restricts to a coaction of the Hopf algebra . We use a variation of Sweedler notation: for we write .

The action by a compact quantum group as above is said to be ergodic if the fixed point subalgebra equals . Then admits a unique -invariant state, which we will denote . It is determined by the following condition

 hX(x)\mathds1=(id⊗hG)(α(x)),x∈\mathnormalL∞(X).

The GNS Hilbert space of will be denoted by . In what follows we shall view as a von Neumann subalgebra of .

Given two compact quantum groups and we say that is a (closed quantum) subgroup of if there exists a surjective Hopf -algebra map . This is equivalent to the existence of an injective normal embedding, respecting the coproducts, of the von Neumann algebra into . Note that in that case we can naturally talk about restricting representations of to , which is the main point of interest for this paper. The subgroup is said to be normal if in addition ([WangFree, Wang2014, centers]), in which case one can define a quotient compact quantum group . Similarly given two discrete quantum groups and we say that is a (closed quantum) subgroup of if we have an injective Hopf -algebra map . This is equivalent to the existence of a surjective -homomorphism from onto , once again intertwining respective coproducts. Most of the above concepts have natural generalizations to arbitrary locally compact quantum groups, which we will briefly use in the beginning of Section 5.1.

For more information on actions of compact quantum groups and related topics we refer the reader to the lecture notes [KennyLectures]. The topics of quantum subgroups are thoroughly covered in [DKSS] and information on normal quantum subgroups, inner automorphisms etc. can be found in [centers] and references therein. We will at some point use the theory of locally compact quantum groups in the sense of Kustermans and Vaes ([KV]).

## 3. Ergodic actions of a compact quantum group on quantum spaces with a discrete component

In this section we show that if a compact quantum group acts ergodically on a “quantum space with a discrete component”, that is on a von Neumann algebra of the form for some , then must be finite (i.e.  is finite-dimensional).

Let be a von Neumann algebra and an ergodic action. Denote by and the GNS maps corresponding to the invariant state on and the Haar measure and consider the isometry

 G:\mathnormalL2(X)⊗\mathnormalL2(X)⟶\mathnormalL2(X)⊗\mathnormalL2(G),

defined by

 ΛX(x)⊗ΛX(y)⟼(ΛX⊗ΛG)(α(y)(x⊗\mathds1)),x,y∈Pol(X).

Since

 (id⊗ωΛX(z),ΛX(y))(G)=(id⊗hG)((\mathds1⊗z∗)α(y)),y,z∈Pol(X),

it follows that

 G∈\mathnormalL∞(X)¯⊗B(\mathnormalL2(X),\mathnormalL2(G)),

so in particular we can consider for the element

 (ω⊗id)(G)∈B(\mathnormalL2(X),\mathnormalL2(G)).

An easy computation shows that

 (ω⊗id)(G)ΛX(x)=ΛX((ω⊗id)(α(x)),x∈\mathnormalL∞(X). (3.1)

For each consider

 Lω=(ω⊗id)(G∗):\mathnormalL2(G)⟶\mathnormalL2(X).

Note that using (3.1) one can easily check that the map is injective.

Let denote the norm-closure of the set of the operators :

 c00(ˆX)=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯{(ω⊗id)(G∗)\vlineω∈\mathnormalL∞(X)∗}∥⋅∥.

The notation is justified by the fact that is in a sense a discrete object, as follows from Lemma 3.3 below.

Applying the construction above to we get a copy of (more precisely the -algebra associated to the left regular representation of on ). In particular, for we shall write .

###### Lemma 3.1.

For , , and the following equalities hold:

 LhX(⋅x)ΛG(a) =hG(S(x(1))a)ΛX(x(0)), (3.2a) LωLχ =Lω⋅χ, (3.2b) L∗hX(⋅y)LhX(⋅x) =hX(y∗x(0))LhG(⋅x(1))=hX(y∗(0)x)L∗hG(⋅y(1)), (3.2c) where ω⋅χ=(ω⊗χ)∘α.
###### Proof.

In order to get (3.2a) we compute

 ⟨ΛX(y)\vlineLhX(⋅x)ΛG(a)⟩ =⟨ΛX(\mathds1)⊗ΛX(y)\vlineG∗ΛX(x)⊗ΛG(a)⟩ =hX(y∗(0)x(0))hG(y∗(1)x(1)S(x(2))a) =⟨ΛX(y)\vlinehG(S(x(1))a)ΛX(x(0))⟩,

where in the fifth equality we used -invariance of .

In order to prove (3.2b) we compute

 L∗χL∗ωΛX(x) =¯¯¯ω(x(0))L∗χΛG(x(1)) =¯¯¯ω(x(0))¯¯¯¯χ(x(1))ΛG(x(2)) =L∗ω⋅χΛX(x).

Finally, in order to prove the first equality of (3.2c) we compute

 L∗hX(⋅y)LhX(⋅x)ΛG(a) =L∗hX(⋅y)hG(S(x(1))a)ΛX(x(0)) =hX(y∗x(0))LhG(⋅x(1))ΛG(a).

The second equality of (3.2c) follows from the first one by taking adjoints of both sides. ∎

###### Corollary 3.2.

For all and we have and . In particular forms a Hilbert -module over .

In the following, we will denote by the finite-dimensional space of matrix coefficients of and write

 Pol(X)ι={(id⊗hG(S−1(a)⋅))α(x)\vlinea∈Pol(G)ι,x∈\mathnormalL∞(X)}

for the corresponding spectral subspace, which is finite-dimensional by [boca, Theorem 17].

Recall that we have

 c00(ˆG)≅⨁ι∈IrrGMnι(C).
###### Lemma 3.3.

There is an isomorphism of -modules

 c00(ˆX)=⨁ι∈IrrGMmι,nι(C) (3.3)

for certain finite (with possibly ) and the obvious Hilbert -module structure.

###### Proof.

It is easy to see that any Hilbert C-module over must be of the form for certain Hilbert spaces , with the Hilbert space underlying the representation . Our aim is to show that the are finite-dimensional.

However, denoting by and the finite-dimensional Hilbert spaces obtained as the images of and under the GNS-construction, it follows that each sends into . Hence and so also is finite-dimensional. ∎

Note that it follows from the above lemma that for each

 Mmι,nι(C)≅{LhX(⋅x)\vlinex∈Pol(X)ι}. (3.4)
###### Theorem 3.4.

Let be an ergodic action of a compact quantum group on . If then .

###### Proof.

Assume that is not of finite dimension, but

 \mathnormalL∞(X)=Mn(C)⊕N.

We will arrive at a contradiction.

Denote by the canonical unital normal -homomorphism obtained by projecting onto the first component. Consider the unital normal -homomorphism defined by

 x⟼xη=(η⊗id)α(x).

Then for define in by the equality

 xη=n∑i,j=1eij⊗xij,

where are the matrix units of . Consider now the convolution operator defined by

 L(id⊗hG)(⋅xη)(ξ⊗ζ)=n∑i,j=1eijξ⊗LhG(⋅xij)ζ,ξ∈Cn,ζ∈\mathnormalL2(G).

We claim that

 ∥∥L(id⊗hG)(⋅xη)∥∥=∥∥LhX(⋅x)∥∥. (3.5)

Indeed, we compute (using (3.2c))

 L∗(id⊗hG)(⋅yη)L(id⊗hG)(⋅xη)=n∑i,j,k,l=1ejiekl⊗L∗hG(⋅yij)LhG(⋅xkl)=n∑j,l,i=1ejl⊗hG(y∗ijxil(1))LhG(⋅xil(2)).

However, we have

 (id⊗hG⊗id)(n∑j,l,i=1ejl⊗y∗ijxil(1)⊗xil(2))=n∑i,j,k,l=1ejiekl⊗hG(y∗ijxkl(1))⊗xkl(2)=(id⊗hG⊗id)((αη(y)∗⊗\mathds1)(id⊗ΔG)αη(x))=(η⊗hG⊗id)(α⊗id)((y∗⊗\mathds1)α(x))=\mathds1⊗(hX⊗id)((y∗⊗\mathds1)α(x))=\mathds1⊗hX(y∗x(0))x(1).

Hence

 L∗(id⊗hG)(⋅yη)L(id⊗hG)(⋅xη)=\mathds1⊗hX(y∗x(0))LhG(⋅x(1))=\mathds1⊗L∗hX(⋅y)LhX(⋅x)

by the identity (3.2c). In particular, (3.5) holds.

We are now ready to obtain our contradiction. Consider the normal linear functionals forming the matrix coefficients of ,

 η(x)=n∑i,j=1ηij(x)eij,

and note that . Then let

 Lη=n∑i,j=1eij⊗Lηij∈Mn(C)⊗c00(ˆX), (3.6)

which is an operator from to . Using (3.2c), we find that for , and we have

However, as , we can find by (3.4) an infinite collection of mutually inequivalent irreducible representations of with corresponding central projections and such that

 LhX(⋅xι)pι=LhX(⋅xι).

Since

 α(Pol(X)ι)⊂Pol(X)ι⊗Pol(G)ι,

we find that

 ∥∥LhX(⋅xι))∥∥=∥∥L(id⊗hG)(⋅(xι)η)∥∥=∥∥(\mathds1⊗pι)L∗η(id⊗LhX(⋅xι))∥∥≤∥∥Lη(\mathds1⊗pι)∥∥∥∥LhX(⋅xι)∥∥.

It follows that for infinitely many , which is a contradiction with (3.3) and (3.6). ∎

## 4. Actions of compact quantum groups on direct sums

In this section we consider an action of a compact quantum group on a direct sum of von Neumann algebras over a certain index set . We study a resulting relation on , classically corresponding to the relation of being in the same orbit of the action.

### 4.1. Actions on direct sums of von Neumann algebras

In the first subsection we consider the most general setup. Let be a compact quantum group, and let be a von Neumann algebra of the following form:

 M=∏i∈IMi,

where is an index set and for each we have a von Neumann algebra . We view each as a (non-unital) subalgebra of and write for the image of in and for the canonical map . Elements of are norm bounded families with each . Given we have where for each . Let be an action of on .

###### Definition 4.1.

Let . We say that is -related to (which we write ) if there exists such that

 (pj⊗id)α(x)≠0.

Define for each the (usually non-unital) -homomorphism ,

 αji(x)=(pj⊗id)α(x),x∈Mi.

Equation (2.1) now takes the form

 ∑k∈I(αjk⊗id)(αki(x))=(id⊗ΔG)(αji(x)),i,j∈I,x∈Mi. (4.1)

Further say that is implemented by a unitary , where is a Hilbert space, if there exists a (unital, normal) faithful representation for which

 (π⊗id)(α(y))=U(π(y)⊗\mathds1)U∗,y∈M. (4.2)

The form of implies that in fact and , where for each the map is a unital representation of . Thus we can in fact write the implementing unitary as a matrix , where . In this picture equation (4.2) can be written as

 (πj⊗id)(αji(x))=Uji(πi(x)⊗\mathds1)U∗ji,i,j∈I,x∈Mi. (4.3)

An action of on can always be implemented. One possible such implementation could be defined by choosing a faithful representation of on a Hilbert space and defining , and . There are many other constructions yielding a unitary implementation (see e.g. [boca, Vaes-implementation]).

###### Lemma 4.2.

Let . Then if and only if , if and only if . Moreover if is implemented by a unitary as above the above conditions are equivalent to the fact that .

###### Proof.

The first two equivalences are obvious. So is the third one, once we note that by (4.3) we have (for )

 (πj⊗id)(αji(\mathds1Mi))=UjiU∗ji.

###### Proposition 4.3.

The relation is symmetric.

###### Proof.

We may and do assume that is implemented by a unitary which is a representation of . If , we have by Lemma 4.2 that . Equivalently, there exist and such that

 (ωξ,η⊗id)(U)≠0.

Note however that belongs to , and moreover, by [mu, Theorem 1.6], it is in the domain of the antipode of and

 (ωξ,η⊗id)(U∗)=S((ωξ,η⊗id)(U))≠0,

where we use the fact that is injective. This means that , so and by Lemma 4.2 we see that . ∎

### 4.2. Actions on direct sums of factors

Easy classical examples (take acting on the set by flipping with and with and write as ) show that in this generality our relation need not be transitive. The lack of transitivity however cannot happen as soon as the “components” are “indecomposable”.

###### Theorem 4.4.

Assume that each is a factor. Then the relation is an equivalence relation. Furthermore, for each equivalence class of the relation the action restricts in an obvious way to an action on . In particular, the projection is invariant under , i.e. .

###### Proof.

Factoriality and normality of the maps involved imply that for any the statement is equivalent to being injective. Thus if further and , then considering equation (4.1) we see that the projection is equal to the sum of projections , dominating the non-zero projection (as by the same token is injective). This shows that and consequently that is transitive.

Finally note that if then there must be such that – otherwise would fail to be injective. This fact together with symmetry established in Proposition 4.3 and transitivity shown above implies that and the proof of the first part is complete.

The second statement is clear.

The last statement follows from the fact that is the unit of the algebra . ∎

We hence from now on assume that each is a factor.

###### Lemma 4.5.

Suppose that is ergodic. Then for any .

###### Proof.

We note that for each equivalence class as in Theorem 4.4. ∎

It is natural to expect that compact quantum groups cannot act on a direct sum of factors in such a way that the equivalence relation introduced above admits an infinite class. The result in full generality remains beyond our reach, but using Theorem 3.4 we can establish it for a direct sum of matrix algebras.

###### Theorem 4.6.

There is no action of a compact quantum group on such that the relation admits an infinite orbit.

###### Proof.

By the last statement in Theorem 4.4 we can assume that the equivalence relation has only one class.

Let be an action as above. Consider a minimal non-zero projection (a minimal projection exists because is a type von Neumann algebra). Then restricts to an ergodic action on (this follows from minimality of ). In particular , which is again of the form , is finite-dimensional by Theorem 3.4. In other words, is finite. Consider then . Assume that there exists . Then

 αji(pip)=(pj⊗id)(α(pip))≤(pj⊗id)(α(p))=pj(p)⊗\mathds1=0.

As by assumption, we obtain by simplicity of and Lemma 4.2 the contradiction . Hence