# On the structure of quantum automorphism groups

###### Abstract.

We compute the -theory of quantum automorphism groups of finite dimensional -algebras in the sense of Wang. The results show in
particular that the -algebras of functions on the quantum permutation groups are pairwise non-isomorphic
for different values of .

Along the way we discuss some general facts regarding torsion in discrete quantum groups. In fact, the duals of quantum automorphism
groups are the most basic examples of discrete quantum groups exhibiting genuine quantum torsion phenomena.

###### 2000 Mathematics Subject Classification:

19D55, 81R50## 1. Introduction

Quantum automorphism groups were introduced by Wang [Wangqsymmetry] in his study of noncommutative symmetries of finite dimensional -algebras.
These quantum groups are quite different from -deformations of compact Lie groups, and interestingly, they appear naturally
in a variety of contexts, including combinatorics and free probability, see for instance [BBCsurvey], [BCSdefinetti].
The -algebraic properties of quantum automorphism groups were studied by Brannan [Brannanquantumautomorphism], revealing
various similarities with free group -algebras.

An interesting subclass of quantum automorphism groups is provided by quantum permutation groups.
Following [BSliberation], we will write for the quantum permutation group on letters. According to the definition of Wang, the quantum
group is the universal compact quantum group acting on the abelian -algebra . If one replaces by
a general finite dimensional -algebra, one has to add the data of a state and restrict to state-preserving actions in the definition of
quantum automorphism groups. Indeed, the choice of state is important in various respects. This is illustrated, for instance, by the work of De Rijdt
and Vander Vennet on monoidal equivalences among quantum automorphism groups [dRV].

The aim of the present paper is to compute the -theory of quantum automorphism groups. Our general strategy follows the ideas in [Voigtbcfo],
which in turn build on methods from the Baum-Connes conjecture, formulated in the language of category theory following
Meyer and Nest [MNtriangulated]. In fact, the main result of [Voigtbcfo] implies rather easily that the
appropriately defined assembly map for duals of quantum automorphism groups is an isomorphism. The main additional ingredient, discussed further below,
is the construction of suitable resolutions, entering the left hand side of the assembly map in the framework of [MNtriangulated].

The reason why this is more tricky than in [Voigtbcfo] is that quantum automorphism groups have torsion. At first sight, the
presence of torsion may appear surprising because these quantum groups behave like free groups in
many respects. Indeed, the way in which torsion enters the picture is different from what happens for classical discrete groups.
Therefore quantum automorphism groups provide an interesting class of examples also from a conceptual point of view. Indeed, a better
understanding of quantum torsion seems crucial in order to go beyond the class of quantum groups studied
in the spirit of the Baum-Connes conjecture so far [MNcompact], [Voigtbcfo], [VVfreeu].
We have therefore included some basic considerations on torsion in discrete quantum groups in this paper.

From our computations discussed below one can actually see rather directly the effect of torsion on the level of -theory. In particular,
the -groups of monoidally equivalent quantum automorphism groups can differ quite significantly due to minor
differences in their torsion structure. Our results also have some direct operator algebraic consequences, most notably, they
imply that the reduced -algebras of functions on quantum permutation groups can be distinguished by -theory.

Let us now explain how the paper is organised. In section 2 we collect some definitions and facts from
the theory of compact quantum groups and fix our notation. Section 3 contains more specific
preliminaries on quantum automorphism groups and their actions. In section LABEL:sectorsion
we collect some basic definitions and facts regarding torsion in discrete quantum groups. In the quantum case, this
is studied most efficiently in terms of ergodic actions of the dual compact quantum groups, and
our setup generalises naturally previous considerations by Meyer and Nest [MNcompact], [Meyerhomalg2].
Finally, section LABEL:seckqaut contains our main results.

Let us conclude with some remarks on notation. We write for the algebra of adjointable operators on a Hilbert module .
Moreover denotes the algebra of compact operators. The closed linear span of a subset of a Banach space is denoted
by . Depending on the context, the symbol denotes either the tensor product of Hilbert spaces,
the spatial tensor product of -algebras, or the exterior tensor product of Hilbert modules.

## 2. Compact quantum groups

In this preliminary section we collect some definitions from the theory of compact quantum groups and
fix our notation. We will mainly follow the conventions in [Voigtbcfo] as far as general quantum group theory is concerned.

Let us start with the following definition.

###### Definition 2.1.

A compact quantum group is given by a unital Hopf -algebra , that is, a unital -algebra together with a unital -homomorphism , called comultiplication, such that

and

For every compact quantum group there exists a Haar state, namely a state satisfying the invariance
conditions for all .
The image of in the GNS-representation of is denoted , and called the reduced
-algebra of functions on . We will write for the GNS-Hilbert space of , and notice that the GNS-representation
of on is faithful.

A unitary representation of on a Hilbert space is a unitary element
such that . In analogy with the classical theory for compact groups,
every unitary representation of a compact quantum group is completely reducible, and irreducible
representations are finite dimensional. We write for the set of equivalence classes of irreducible unitary
representations of . The linear span of matrix coefficients of all unitary representations of forms a dense Hopf -algebra
of .

The full -algebra of functions on is the universal -completion of . It admits a comultiplication
as well, satisfying the density conditions in definition 2.1.
The quantum group can be equivalently described in terms of or , or in fact,
using . One says that is coamenable if the canonical quotient map
is an isomorphism. In this case we will simply write again for this -algebra. By slight abuse of notation,
we will also write if a statement holds for both and .

The regular representation of is the representation of on corresponding to the
multiplicative unitary determined by

where is the image of under the GNS-map. The comultiplication of can be recovered from by the formula

One defines the algebra of functions on the dual discrete quantum group by

together with the comultiplication

for , where . We remark that
there is no need to distinguish between and in the discrete case.

Since we are following the conventions of Kustermans and Vaes [KVLCQG], there is a flip map built into the above definition of ,
so that the comultiplication of corresponds to the opposite multiplication of . This is a natural choice
in various contexts, but it is slightly inconvenient when it comes to Takesaki-Takai duality. We will
write for , that is, for the Hopf -algebra equipped with the
opposite comultiplication , where denotes the flip map. By slight abuse
of terminology, we shall refer to both and as the dual quantum group of , but in the sequel
we will always work with instead of . According to Pontrjagin duality, the double dual of in either of the two conventions
is canonically isomorphic to .

An action of a compact quantum group on a -algebra is a coaction of on , that is, an injective
nondegenerate -homomorphism
such that and .
In a similar way one defines actions of discrete quantum groups, or in fact arbitrary locally compact quantum groups. We will
call a -algebra equipped with a coaction of a --algebra. Moreover we write - for the
category of all --algebras and equivariant -homomorphisms.

The reduced crossed product of a --algebra is the -algebra

The crossed product is equipped with a canonical dual action of , which turns it into a --algebra. Moreover, one has the following analogue of the Takesaki-Takai duality theorem [BSUM].

###### Theorem 2.2.

Let be a regular locally compact quantum group and let be a --algebra. Then there is a natural isomorphism

of --algebras.

We will use Takesaki-Takai duality only for discrete and compact quantum groups, and in this setting regularity is automatic. At some points we will also use the full crossed product of a --algebra , and we refer to [NVpoincare] for a review of its definition in terms of its universal property for covariant representations.

## 3. Quantum automorphism groups

In this section we review some basic definitions and results on quantum automorphism groups of finite dimensional
-algebras and fix our notation. We refer to [Wangqsymmetry], [Banicageneric], [Banicafusscatalan] for more background on quantum automorphism groups.

Let us start with the definition of the quantum automorphism group of a finite dimensional -algebra ,
compare [Wangqsymmetry] [Banicageneric]. If denotes the multiplication map then a faithful
state on is called a -form for if with respect to the Hilbert space
structures on and implemented by the GNS-constructions for and , respectively.

###### Definition 3.1.

Let be a finite dimensional -algebra and let be a -form on for some . The quantum automorphism
group is the universal compact quantum group acting on such that is preserved.

That is, if is any compact quantum group together with a coaction
then there exists a unique morphism of quantum groups such that the diagram