Quantum automorphisms of twisted group algebras and free hypergeometric laws

Teodor Banica, Julien Bichon, Stephen Curran

Introduction

The notion of quantum automorphism group was introduced about 10 years ago, in Wang’s paper . The idea is as follows:

Wang proved that this space $X$ has a quantum automorphism group $G_{aut}(X)$ . In algebraic terms, the result is that there is a universal Hopf algebra $A_{aut}(A)$ , coacting on $A$ , and leaving the canonical trace invariant.

The quantum automorphism groups, and their quantum subgroups, were systematically investigated in the last years. See e.g. , , , , .

In this paper we establish a general isomorphism result, of the following type:

In other words, the above general algebraic result leads in this case to a non-trivial relation between quantum rotations and quantum permutations!

We will further investigate this phenomenon, by using concepts from Voiculescu’s free probability theory . The idea is that, probabilistically speaking, the above result tells us that the variables $u_{ij}^{2}\in A_{o}(n)$ have the same law as the following variables:

Here $u_{ij}$ and $p_{ia,jb}$ are respectively the standard generators of $A_{o}(n)$ and $A_{s}(n^{2})$ .

The point now is that the variables $u_{ij}$ can be regarded as being “free hyperspherical variables”, and the variables $X_{ij}$ can be regarded as being (rescaled) “free hypergeometric variables”. So, what we have here is a new free probability formula, which at $n=\infty$ corresponds to the well-known relation between the semicircle law, and the free Poisson law. We will make several comments on this result, including a further investigation of the free hypergeometric laws, for more general values of the parameters.

It is a pleasure to thank B. Collins and R. Speicher for several useful discussions, prior to the work leading to the present article. The work of T.B. and J.B. was supported by the ANR grant “Galoisint”.

The twisting result

We let $A_{aut}(A)$ be the universal Hopf algebra coacting on the algebra $A$ , such that the canonical trace $tr$ is invariant. This construction is the $\phi=tr$ particular case of Wang’s general construction $A_{aut}(A,\phi)$ , from . Regarding the choice $\phi=tr$ , see .

The comultiplication, counit and antipode are given by the following formulae:

Once again, this follows from a direct verification. Note that by using cocycle identities we obtain $\sigma(g,g^{-1})=\sigma(g^{-1},g)$ , needed in the proof. ∎

Given a left 2-cocycle $\sigma$ on $H$ , one can form the 2-cocycle twist $H^{\sigma}$ as follows. As a coalgebra, $H^{\sigma}=H$ , and an element $x\in H$ , when considered in $H^{\sigma}$ , is denoted $[x]$ . The product in $H^{\sigma}$ is defined, in Sweedler notation, by:

Note that the cocycle condition ensures the fact that we have indeed a Hopf algebra. For the formula of the antipode, that we will not need here, see .

Note that the coalgebra isomorphism $H\to H^{\sigma}$ given by $x\to[x]$ commutes with the respective Haar integrals (as soon as $H$ has a Haar integral, of course).

We are now in position to state and prove our main theorem.

Associated with any $2$ -cocycle are the following quantities:

With this notation, we have the following technical reformulation of Theorem 1.3.

If $G$ is a finite group and $\sigma$ is a $2$ -cocycle on $G$ , then

This is indeed just a technical reformulation of Theorem 1.3. ∎

Rotations and permutations

In this section we discuss some concrete consequences of the general results established in the previous section. These will concern Wang’s quantum permutation groups .

Consider indeed the following linear map:

It is routine to check that both $\psi,\psi^{\prime}$ are morphisms of algebras, and that these maps are inverse to each other. In particular, $\psi$ is an isomorphism of algebras, as stated. ∎

Consider the universal coactions on the two algebras in the statement:

In terms of the standard bases, these coactions are given by:

By comparing with the formula of $\alpha$ , we obtain the isomorphism in the statement. ∎

Consider now the Wang algebras $A_{o}(n)$ and $A_{s}(n^{2})$ , with standard generators denoted $(u_{ij})_{i,j=1,\ldots,n}$ and $(p_{ia,jb})_{i,j,a,b=1,\ldots,n}$ . That is, $u=(u_{ij})$ is the universal $n\times n$ orthogonal matrix, and $p=(p_{ia,jb})$ is the universal $n^{2}\times n^{2}$ magic unitary matrix. See .

We recall that we have canonical identifications, as follows:

Here the projective version of a pair $(A,u)$ is by definition the pair $(PA,v)$ , where $v=u\otimes\bar{u}$ and $PA=<v_{ij}>$ . For full details regarding the above equalities, see .

defines a coalgebra isomorphism $PA_{o}(n)\to A_{s}(n^{2})$ , commuting with the Haar integrals.

This follows from the general isomorphism results in Theorem 1.3 and Proposition 1.4, by combining them with the various isomorphisms from the lemmas above. ∎

The following two algebras are isomorphic, via $u_{ij}^{2}\to X_{ij}$ :

The algebra generated by the variables $u_{ij}^{2}\in A_{o}(n)$ .

The algebra generated by $X_{ij}=\frac{1}{n}\sum_{a,b=1}^{n}p_{ia,jb}\in A_{s}(n^{2})$

Free hypergeometric laws

With this notation, we know from Theorem 2.5 that the variables $u_{ij}^{2}\in A_{o}(n)$ have the same joint law as the variables $X_{ij}\in A_{s}(n^{2})$ . In this section we will put this result into a more general framework, related to Voiculescu’s free probability theory .

Let us begin by giving an independent proof for the above equality of distributions. We use the Weingarten formula . We recall that the representation-theoretic sets of partitions for the algebras $A=A_{o}(n),A_{s}(n^{2})$ are respectively the set of noncrossing pairings $NC_{2}(2k)$ , and the set of noncrossing partitions $NC(k)$ . Associated to each of these sets are the Gram matrix $G_{m}(\pi,\sigma)=m^{|\pi\vee\sigma|}$ and the Weingarten matrix $W_{m}=G_{m}^{-1}$ , where $\vee$ is the join operation, and $m=n,n^{2}$ respectively. The Haar functional of $A$ can be computed explicitely in terms of $W_{m}$ . For full details here, see .

The Weingarten matrices of $A_{o}(n)$ and $A_{s}(n^{2})$ are related by

We use the following general formula, due to Kodiyalam and Sunder :

See also . Now in terms of Gram matrices, we obtain:

By taking the inverse, this gives the formula in the statement. ∎

has the same law as the family of variables $\{u_{ij}^{2}\}\subset A_{o}(n)$ .

We use the Weingarten formula, which gives:

By Lemma 3.1 the terms on the right are equal, and this gives the equality of laws for the individual variables. For the general statement, the proof is similar. ∎

The variables $X_{ij}$ appearing in Theorem 3.2 have the following generalization.

is called free hypergeometric, of parameters $(n,m,N)$ .

The terminology here comes from the fact that the variable $X^{\prime}(n,m,N)$ , defined as above, but over the algebra $C(S_{n})$ , follows a hypergeometric law of parameters $(n,m,N)$ .

In general, the free hypergeometric laws seem to be quite difficult to compute. A first result in this direction, heavily relying on a result recently obtained in , is as follows.

The moments of $X(n,n,n^{2})$ are given by

where $q\in[-1,0)$ is given by $q+q^{-1}=-n$ .

First, $X(n,n,n^{2})/n$ is the variable $X_{ij}$ appearing by Theorem 3.2, having the same law as the variable $u_{ij}^{2}\in A_{o}(n)$ . Now it is known from that $A_{o}(n)$ is monoidally equivalent to $C(SU_{q}(2))$ , and, as explained in , one can use this fact for modelling $u_{ij}\in A_{o}(n)$ by a certain variable over $SU_{q}(2)$ . This latter variable can be studied by using advanced calculus methods, and this leads to the above formula. See . ∎

As a first observation, the above result, or rather a version of it, namely Theorem 5.3 in , shows that the variables $X(n,n,n^{2})$ superconverge with $n\to\infty$ . For more about the superconvergence phenomenon in free probability, see Bercovici and Voiculescu .

The second result, that we would like to present now, is an exploration of the basic asymptotic properties of the free hypergeometric laws.

The free hypergeometric laws have the following properties:

Let $n,m,N\to\infty$ , with $\frac{nm}{N}\to\lambda\in(0,\infty)$ . Then the law of $X(n,m,N)$ converges to the free Poisson law of parameter $\lambda$ .

Let $n,m,N\to\infty$ , with $\frac{n}{N}\to\nu\in(0,1)$ and $\frac{m}{N}\to 0$ . Then the law of $S(n,m,N)=(X(n,m,N)-m\nu)/\sqrt{m\nu(1-\nu)}$ converges to a $(0,1)$ -semicircle law.

(1) From the Weingarten formula, we have:

Now, as explained for instance in , we have:

Thus the $p$ -th moment of $X(n,m,N)$ converges to $\sum_{\pi\in NC(p)}\lambda^{|\pi|}$ , which is the $p$ -th moment of the free Poisson distribution with parameter $\lambda$ , and we are done.

(2) We need to show that the free cumulants satisfy:

The case $p=1$ is trivial, so suppose $p\geq 2$ . We have:

On the other hand, from the Weingarten formula, we have:

We use now the following standard identity:

This gives the following formula for the cumulants:

It follows that for $p\geq 3$ we have, as desired:

As for the remaining case $p=2$ , here we have:

Concluding remarks

We have seen in this paper that Wang’s quantum automorphism groups in are subject to some general twisting results, and that these results are of relevance in the general context of Voiculescu’s free probability theory . Several questions appear:

What is the most general twisting result for quantum automorphism groups? An answer here is $A_{aut}(H_{\sigma})\simeq A_{aut}(H)^{\sigma}$ , for any finite dimensional Hopf algebra $H$ . This result, whose proof is much more technical, will be discussed somewhere else.

Can one include the monoidal equivalence used in into the above considerations? The point is that the variable $u_{ij}\in A_{o}(n)$ has the same law as a certain variable over $SU_{q}(2)$ , where $q+q^{-1}=-n$ , so the result in can be probably stated and proved by using $A_{s}(n^{2})$ and $SU_{q}(2)$ only. However, it is not clear how to do so.

Are there any other free hypergeometric laws, that can be explicitely computed by using $A_{o}(F)$ ? In principle the answer here is no, first because of the concluding remarks in , and second, because of the “no-atoms” results of Voigt in .

Do we have superconvergence, in the sense of Bercovici and Voiculescu , to the limiting distributions in Theorem 3.5? Note that in the case $n=m=\sqrt{N}$ the superconvergence appears indeed, as explained after Theorem 3.4.

Finally, let us point out the fact that, as explained in , the free hyperspherical laws appear in connection with the study of a number of interesting “noncommutative spaces”, such as the free and half-liberated spheres, or projective spaces. We do not know yet if the relation with the free hypergeometric laws, that we found in this paper, can be of help here, but we intend to come back to this question in some future work.