De Finetti theorems for easy quantum groups

Introduction

In the study of probabilistic symmetries, the classicalgroups $S_{n}$ and $O_{n}$ play central roles. De Finetti’s fundamental theorem states that an infinite sequence of random variables whose joint distribution is invariant under finite permutations must be conditionally independent and identically distributed. In fre1 , Freedman considered sequences of real-valued random variables whose joint distribution is invariant under orthogonal transformations, and proved that any infinite sequence with this property must form a conditionally independent Gaussian family with mean zero and common variance. Although these results fail for finite sequences, approximation results may still be obtained (see df1 , df2 ). For a thorough treatment of probabilistic symmetries, the reader is referred to the recent text of Kallenberg kal .

The free analogues $S_{n}^{+}$ and $O_{n}^{+}$ of the permutation and orthogonal groups were constructed by Wang in wang1 , wang2 . These are compact quantum groups in the sense of Woronowicz wor1 . In ksp , Köstler and Speicher discovered that de Finetti’s theorem has a natural free analogue: an infinite sequence of noncommutative random variables has a joint distribution which is invariant under “quantum permutations” coming from $S_{n}^{+}$ if and only if the variables are freely independent and identically distributed with amalgamation, that is, with respect to a conditional expectation. This was further studied in cur1 , where this result was extended to more general sequences and an approximation result was given for finite sequences. The free analogue of Freedman’s result was obtained in cur2 , where it was shown that an infinite sequence of self-adjoint noncommutative random variables has a joint distribution which is invariant under “quantum orthogonal transformations” if and only if the variables form an operator-valued free semicircular family with mean zero and common variance.

In this paper, we present a unified approach to de Finetti theorems by using the “easiness” formalism from bsp . Stated roughly, a quantum group $S_{n}\subset G\subset O_{n}^{+}$ is called easy if its tensor category is spanned by certain partitions coming from the tensor category of $S_{n}$. This might look, of course, to be a quite technical condition. However, we feel that this provides a good framework for understanding certain probabilistic and representation theory aspects of orthogonal quantum groups. There are 14 natural examples of easy quantum groups, listed as follows: {longlist}[(3)]

Groups: $O_{n},S_{n},H_{n},B_{n},S_{n}^{\prime},B_{n}^{\prime}$.

Free versions: $O_{n}^{+},S_{n}^{+},H_{n}^{+},B_{n}^{+},S_{n}^{\prime+},B_{n}^{\prime+}$.

As explained in bsp , bcs1 , our motivating belief is that “any result which holds for $S_{n},O_{n}$ should have a suitable extension to all easy quantum groups.” This is, of course, a quite vague statement, whose target is formed by several results at the borderline of representation theory and probability. This paper represents the first application of this philosophy.

If $G$ is an easy quantum group, there is a natural notion of $G$-invariance for a sequence of noncommutative random variables, which agrees with the usual definition when $G$ is a classical group. Our main result is the following de Finetti type theorem, which characterizes the joint distributions of infinite $G$-invariant sequences for the 10 natural easy quantum groups discussed above.

The notion of half-independence, appearing in (2) above, will be introduced in Section 2. The basic example of a half-independent family of noncommutative random variables is $(x_{i})_{i\in I}$,

The paper is organized as follows. Section 1 contains preliminaries. Here we collect the basic notions from the combinatorial theory of classical and free probability. We also recall some basic notions and results from bsp about the class of “easy” quantum groups. In Section 2, we introduce half-independence and develop its basic combinatorial theory. In Section 3, we recall the Weingarten formula from bsp for computing integrals on easy quantum groups, and give a new estimate on the asymptotic behavior of these integrals. This will be essential to the proofs of our main results, and we believe that this estimate will also find applications to other problems involving easy quantum groups. In Section 4, we define quantum invariance for finite sequences, prove a converse to Theorem 1, and give approximate de Finetti type results. Section 5 contains the proof of Theorem 1, and a discussion of the situation for unbounded random variables in the classical and half-liberated cases. Section 6 contains concluding remarks.

Background and notation

We begin by recalling the basic notions of noncommutative probability spaces and distributions of random variables. For further details, see the texts vdn , ns .

A W∗-probability space $(M,\varphi)$ is a von Neumann algebra $M$ together with a faithful normal state $\varphi$. We will not assume that $\varphi$ is a trace.

Let $(\Omega,\Sigma,\mu)$ be a (classical) probability space. {longlist}[(2)]

Note that the joint distribution of $(x_{i})_{i\in I}$ is determined by the collection of joint moments

In the classical de Finetti’s theorem, the independence which occurs is only after conditioning. Likewise the free de Finetti’s theorem is a statement about freeness with amalgamation. Both of these concepts may be expressed in terms of operator-valued probability spaces, which we now recall.

An operator-valued probability space $(\mathcal{A},E\dvtx\mathcal{A}\to\mathcal{B})$ consists of a unital algebra $\mathcal{A}$, a subalgebra $1\in\mathcal{B}\subset\mathcal{A}$, and a conditional expectation $E\dvtx\mathcal{A}\to\mathcal{B}$, that is, $E$ is a linear map such that $E=1$ and

for all $b_{1},b_{2}\in\mathcal{B}$ and $a\in\mathcal{A}$.

Let $(\mathcal{A},E\dvtx\mathcal{A}\to\mathcal{B})$ be an operator-valued probability space, and let $(x_{i})_{i\in I}$ be a family in $\mathcal{A}$. The $B$-valued joint distribution of the family $(x_{i})_{i\in I}$ is the linear map $E_{x}\dvtx\mathcal{B}\langle t_{i}\dvtx i\in I\rangle\to\mathcal{B}$ defined by

Observe that the joint distribution is determined by the $\mathcal{B}$-valued joint moments

for $b_{0},\ldots,b_{k}\in\mathcal{B}$ and $i_{1},\ldots,i_{k}\in I$. Observe that if $\mathcal{B}$ commutes with the variables $(x_{i})_{i\in I}$, then

so that the $\mathcal{B}$-valued joint distribution is determined simply by the collection of moments $E[x_{i_{1}}\cdots x_{i_{k}}]$ for $i_{1},\ldots,i_{k}\in I$.

Let $(x_{i})_{i\in I}$ be a family in the operator-valued probability space $(\mathcal{A},E\dvtx\mathcal{A}\to\mathcal{B})$. {longlist}[(2)]

If the algebra generated by $\mathcal{B}$ and $\{x_{i}\dvtx i\in I\}$ is commutative, then the variables are called conditionally independent given $B$ if

whenever $i_{1},\ldots,i_{k}$ are distinct and $p_{1},\ldots,p_{k}$ are polynomials in $\mathcal{B}\langle t\rangle$.

The variables $(x_{i})_{i\in I}$ are called free with amalgamation over $\mathcal{B}$, or free with respect to $E$, if

whenever $i_{1},\ldots,i_{k}\in I$ are such that $i_{l}\neq i_{l+1}$ for $1\leq l<k$, and $p_{1},\ldots,p_{k}\in\mathcal{B}\langle t\rangle$ are such that $E[p_{l}(x_{i_{l}})]=0$ for $1\leq l\leq k$.

Voiculescu first defined freeness with amalgamation, and developed its basic theory in voi . Conditional independence and freeness with amalgamation also have rich combinatorial theories, which we now recall. In the free case this is due to Speicher sp2 ; see also ns .

A partition $\pi$ of a set $S$ is a collection of disjoint, nonempty sets $V_{1},\ldots,V_{r}$ such that $V_{1}\cup\cdots\cup V_{r}=S$. $V_{1},\ldots,V_{r}$ are called the blocks of $\pi$, and we set $|\pi|=r$. The collection of partitions of $S$ will be denoted $P(S)$, or in the case that $S=\{1,\ldots,k\}$ by $P(k)$.

Given $\pi,\sigma\in P(S)$, we say that $\pi\leq\sigma$ if each block of $\pi$ is contained in a block of $\sigma$. There is a least element of $P(S)$ which is larger than both $\pi$ and $\sigma$, which we denote by $\pi\vee\sigma$.

If $S$ is ordered, we say that $\pi\in P(S)$ is noncrossing if whenever $V,W$ are blocks of $\pi$ and $s_{1}<t_{1}<s_{2}<t_{2}$ are such that $s_{1},s_{2}\in V$ and $t_{1},t_{2}\in W$, then $V=W$. The set of noncrossing partitions of $S$ is denoted by $\mathit{NC}(S)$, or by $\mathit{NC}(k)$ in the case that $S=\{1,\ldots,k\}$.

The noncrossing partitions can also be defined recursively, a partition $\pi\in P(S)$ is noncrossing if and only if it has a block $V$ which is an interval, such that $\pi\setminus V$ is a noncrossing partition of $S\setminus V$.

Given $i_{1},\ldots,i_{k}$ in some index set $I$, we denote by $\ker\mathbf{i}$ the element of $P(k)$ whose blocks are the equivalence classes of the relation

Note that if $\pi\in P(k)$, then $\pi\leq\ker\mathbf{i}$ is equivalent to the condition that whenever $s$ and $t$ are in the same block of $\pi$, $i_{s}$ must equal $i_{t}$.

Let $(\mathcal{A},E\dvtx\mathcal{A}\to\mathcal{B})$ be an operator-valued probability space. {longlist}[(2)]

A $\mathcal{B}$-functional is a $n$-linear map $\rho\dvtx\mathcal{A}^{n}\to\mathcal{B}$ such that

for all $b_{0},\ldots,b_{n}\in\mathcal{B}$ and $a_{1},\ldots,a_{n}$. Equivalently, $\rho$ is a linear map from $\mathcal{A}^{\otimes_{\mathcal{B}}n}$ to $\mathcal{B}$, where the tensor product is taken with respect to the natural $\mathcal{B}-\mathcal{B}$-bimodule structure on $\mathcal{A}$.

where if $V=(i_{1}<\cdots<i_{s})$ is a block of $\pi$ then

If $\mathcal{B}$ is noncommutative, there is no natural order in which to compute the product appearing in the above formula for $\rho^{(\pi)}$. However, the nesting property of noncrossing partitions allows for a natural definition of $\rho^{(\pi)}$ for $\pi\in\mathit{NC}(n)$, which we now recall from sp2 .

If $\pi=1_{n}$ is the partition containing only one block, define $\rho^{(\pi)}=\rho^{(n)}$.

Otherwise, let $V=\{l+1,\ldots,l+s\}$ be an interval of $\pi$ and define

then the corresponding $\rho^{(\pi)}$ is given by

Let $(\mathcal{A},E\dvtx\mathcal{A}\to\mathcal{B})$ be an operator-valued probability space, and let $(x_{i})_{i\in I}$ be a family of random variables in $\mathcal{A}$. {longlist}[(2)]

The operator-valued classical cumulants $c_{E}^{(k)}\dvtx\mathcal{A}^{k}\to\mathcal{B}$ are the $\mathcal{B}$-functionals defined by the classical moment-cumulant formula

Note that the right-hand side of the equation is equal to $c_{E}^{(n)}[a_{1},\ldots,a_{n}]$ plus lower order terms, and hence $c_{E}^{(n)}$ can be solved for recursively.

The operator-valued free cumulants $\kappa_{E}^{(k)}\dvtx\mathcal{A}^{k}\to\mathcal{B}$ are the $\mathcal{B}$-functionals defined by the free moment-cumulant formula

As above, this equation can be solved recursively for $\kappa_{E}^{(n)}$.

While the definitions of conditional independence and freeness with amalgamation given above appear at first to be quite different, they have very similar expressions in terms of cumulants. In the free case, the following theorem is due to Speicher sp2 .

Let $(\mathcal{A},E\dvtx\mathcal{A}\to\mathcal{B})$ be an operator-valued probability space, and $(x_{i})_{i\in I}$ a family of random variables in $\mathcal{A}$. {longlist}[(2)]

If the algebra generated by $\mathcal{B}$ and $(x_{i})_{i\in I}$ is commutative, then the variables are conditionally independent given $B$ if and only if

whenever there are $1\leq k,l\leq n$ such that $i_{k}\neq i_{l}$.

The variables are free with amalgamation over $\mathcal{B}$ if and only if

whenever there are $1\leq k,l\leq n$ such that $i_{k}\neq i_{l}$.

Note that the condition in (1) is equivalent to the statement that if $\pi\in P(n)$, then

unless $\pi\leq\ker\mathbf{i}$, and likewise in (2) for $\pi\in\mathit{NC}(n)$. Stronger characterizations of the joint distribution of $(x_{i})_{i\in I}$ can be given by specifying what types of partitions may contribute nonzero cumulants.

Let $(\mathcal{A},E\dvtx\mathcal{A}\to\mathcal{B})$ be an operator-valued probability space, and let $(x_{i})_{i\in I}$ be a family of random variables in $\mathcal{A}$. {longlist}[(2)]

Suppose that $\mathcal{B}$ and $(x_{i})_{i\in I}$ generate a commutative algebra. The $\mathcal{B}$-valued joint distribution of $(x_{i})_{i\in I}$ has the property corresponding to $D$ in the table below if and only if for any $\pi\in P(n)$

unless $\pi\in D(n)$ and $\pi\leq\ker\mathbf{i}$.

The $\mathcal{B}$-valued joint distribution of $(x_{i})_{i\in I}$ has the property corresponding to $D$ in the the table below if and only if for any $\pi\in\mathit{NC}(n)$

unless $\pi\in D(n)$ and $\pi\leq\ker\mathbf{i}$.

These results are well known. In the classical case, note that the results for $P_{2},P_{b}$ are equivalent to the Wick formula for computing moments of independent Gaussian families. In the free case, see sp2 , ns .

It is clear from the definitions that the classical and free cumulants can be solved for from the joint moments. In fact, a combinatorial formula for the cumulants in terms of the moments can be given. First we recall the definition of the Möbius function on a partially ordered set.

Let $(\mathcal{A},E\dvtx\mathcal{A}\to\mathcal{B})$ be an operator-valued probability space, and let $(x_{i})_{i\in I}$ be a family of random variables. Define the $\mathcal{B}$-valued moment functionals $E^{(n)}$ by

Suppose that $\mathcal{B}$ is commutative. Then for any $\sigma\in P(n)$ and $a_{1},\ldots,\allowbreak a_{n}\in\mathcal{A}$, we have

For any $\sigma\in\mathit{NC}(n)$ and $a_{1},\ldots,a_{n}\in\mathcal{A}$, we have

This follows from the Möbius inversion formula; see sp2 , ns .

Easy quantum groups

We will now briefly recall some notions and results from bsp .

Consider a compact group $G\subset O_{n}$. By the Stone–Weierstrauss theorem, $C(G)$ is generated by the $n^{2}$ coordinate functions $u_{ij}$ sending a matrix in $G$ to its $(i,j)$ entry. The structure of $G$ as a compact group is captured by the commutative Hopf C∗-algebra $C(G)$ together with comultiplication, counit and antipode determined by

Dropping the condition of commutativity, we obtain the following definition, adapted from the fundamental paper of Woronowicz wor1 .

An orthogonal Hopf algebra is a unital C∗-algebra $A$ generated by $n^{2}$ self-adjoint elements $u_{ij}$, such that the following conditions hold: {longlist}[(3)]

The inverse of $u=(u_{ij})\in M_{n}(A)$ is the transpose $u^{t}=(u_{ji})$.

$\Delta(u_{ij})=\sum_{k}u_{ik}\otimes u_{kj}$ determines a morphism $\Delta\dvtx A\to A\otimes A$.

$S(u_{ij})=u_{ji}$ defines a morphism $S\dvtx A\to A^{op}$.

It follows from the definitions that $\Delta,\varepsilon,S$ satisfy the usual Hopf algebra axioms. If $A$ is an orthogonal Hopf algebra, we use the heuristic formula “$A=C(G)$,” where $G$ is an compact orthogonal quantum group. Of course if $A$ is noncommutative, then $G$ cannot exist as a concrete object, and all statements about $G$ must be interpreted in terms of the Hopf algebra $A$.

The following two examples, constructed by Wang in wang1 , wang2 , are fundamental to our considerations.

$A_{o}(n)$ is the universal C∗-algebra generated by $n^{2}$ self-adjoint elements $u_{ij}$, such that $u=(u_{ij})\in M_{n}(A_{o}(n))$ is orthogonal.

$A_{s}(n)$ is the universal C∗-algebra generated by $n^{2}$ projections $u_{ij}$, such that the sum along any row or column of $u=(u_{ij})\in M_{n}(A_{s}(n))$ is the identity.

As discussed above, we use the notation $A_{o}(n)=C(O_{n}^{+})$, $A_{s}(n)=C(S_{n}^{+})$, and call $O_{n}^{+}$ and $S_{n}^{+}$ the free orthogonal group and free permutation group, respectively.

Here the $\delta$ symbol appearing on the right-hand side is 1 when the indices “fit,” that is, if each block of $\pi$ contains equal indices, and 0 otherwise.

It follows from the above discussion that $\operatorname{Hom}(u^{\otimes k},u^{\otimes l})$ consists of certain linear combinations of the operators $T_{\pi}$, with $\pi\in P(k,l)$. We call $G$ “easy” if these spaces are spanned by partitions.

There are four natural examples of classical groups which are easy:

There is a one-to-one correspondence between classical easy groups and free quantum groups, which on a combinatorial level corresponds to restricting to noncrossing partitions:

There are also free versions of $S_{n}^{\prime},B_{n}^{\prime}$, constructed in bsp .

In general, the class of easy quantum groups appears to be quite rigid (see bcs1 for a discussion here). However, two more examples can be obtained as “half-liberations.” The idea is that instead of removing the commutativity relations from the generators $u_{ij}$ of $C(G)$ for a classical easy group $G$, which would produce $C(G^{+})$, we instead require that the the generators “half-commute,” that is, $abc=cba$ for $a,b,c\in\{u_{ij}\}$. More precisely, we define $C(G^{*})=C(G^{+})/I$, where $I$ is the ideal generated by the relations $abc=cba$ for $a,b,c\in\{u_{ij}\}$. For $G=S_{n},S_{n}^{\prime},B_{n},B_{n}^{\prime}$ we have $G^{*}=G$, however for $O_{n},H_{n}$, we obtain new quantum groups $O_{n}^{*},H_{n}^{*}$. The corresponding partition categories $E_{2},E_{h}$ consist of all pair partitions, respectively all partitions, which are balanced in the sense that each block contains as many odd as even legs.

Half independence

In this section, we introduce a new kind of independence which appears in the de Finetti theorems for the half-liberated quantum groups $H^{*}$ and $O^{*}$. To define this notion, we require that the variables have a certain degree of commutativity.

Let $(x_{i})_{i\in I}$ be a family of noncommutative random variables. We say that the variables half-commute if

Observe that if $(x_{i})_{i\in I}$ half-commute, then in particular $x_{i}^{2}$ commutes with $x_{j}$ for any $i,j\in I$.

Let $(\mathcal{A},E\dvtx\mathcal{A}\to\mathcal{B})$ be an operator-valued probability space, and suppose that $\mathcal{B}$ is contained in the center of $\mathcal{A}$. Let $(x_{i})_{i\in I}$ be a family of random variables in $\mathcal{A}$ which half-commute. We say that $(x_{i})_{i\in I}$ are conditionally half-independent given $\mathcal{B}$, or half-independent with respect to $E$, if the following conditions are satisfied: {longlist}[(2)]

The variables $(x_{i}^{2})_{i\in I}$ are conditionally independent given $\mathcal{B}$.

For any $i_{1},\ldots,i_{k}\in I$, we have

As a first remark, we note that half-independence is defined only between random variables and not at the level of algebras, in contrast with classical and free independence. In fact, it is known from sp1 there are no other good notions of independence between unital algebras other than classical and free.

The conditions may appear at first to be somewhat artificial, but are motivated by the following natural example.

Let $(\Omega,\Sigma,\mu)$ be a (classical) probability space, and let $L(\mu)$ denote the algebra of complex-valued random variables on $\Omega$ with finite moments of all orders. {longlist}[(2)]

Let $(\xi_{i})_{i\in I}$ be a family of independent random variables in $L(\mu)$. Suppose that for each $i\in I$, the distribution of $\xi_{i}$ is such that

A simple computation shows that the variables $(x_{i})_{i\in I}$ half-commute. Since

Observe also that the distribution of $x_{i}$ is equal to that of $(\xi_{i}\overline{\xi_{i}})^{1/2}$, where the square root is chosen such that the distribution is even. We call this the squeezed version of the complex distribution $\xi_{i}$ (cf. bsp ).

Of particular interest is the case that the $(\xi_{i})_{i\in I}$ have complex Gaussian distributions. Here the distribution of $x_{i}$ is the squeezed version of the complex Gaussian $\xi_{i}$, which is a symmetrized Rayleigh distribution.

We will show in Proposition 2.8 below that any half-independent family can be modeled as in the example above. First, we will show that, as for classical and free independence, the joint distribution of a family of half-independent random variables $(x_{i})_{i\in I}$ is determined by the distributions of $x_{i}$ for $i\in I$. It is convenient to first introduce the following family of permutations which are related to the half-commutation relation.

We say that a permutation $\omega\in S_{n}$ preserves parity if $\omega(i)\equiv i~{}(\operatorname{mod}2)$ for $1\leq i\leq n$.

The collection of parity preserving partitions in $S_{n}$ clearly form a subgroup, which is simply $S(\{1,3,\ldots\})\times S(\{2,4,\ldots\})$. Moreover, this subgroup is generated by the transpositions $(i\;i+2)$ for $1\leq i\leq n-2$. It follows that if $(x_{i})_{i\in I}$ half-commute, then

whenever $\omega\in S_{n}$ preserves parity.

Let $(\mathcal{A},E\dvtx\mathcal{A}\to\mathcal{B})$ be an operator-valued probability space such that $\mathcal{B}$ is contained in the center of $\mathcal{A}$. Suppose that $(x_{i})_{i\in I}$ is a family of random variables in $\mathcal{A}$ which are conditionally half-independent given $\mathcal{B}$. Then the $\mathcal{B}$-valued joint distribution of $(x_{i})_{i\in I}$ is uniquely determined by the $\mathcal{B}$-valued distributions of $x_{i}$ for $i\in I$.

Let $i_{1},\ldots,i_{k}\in I$. We know that

unless we have that for each $i\in I$, the set of $1\leq j\leq k$ such that $i_{j}=i$ has as many odd as even elements. So suppose that this the case. By the remark above, we know that $x_{i_{1}}\cdots x_{i_{k}}=x_{i_{\omega(1)}}\cdots x_{i_{\omega(k)}}$ whenever $\omega\in S_{k}$ is parity preserving. With an appropriate choose of $\omega$, it follows that

Let $(X_{i})_{i\in I}$ be a family of independent random variables such that $X_{i}$ has the same distribution as $x_{i}$. Let $(U_{i})_{i\in I}$ be a family of independent Haar unitary random variables which are independent from $(X_{i})_{i\in I}$, and let $\xi_{i}=U_{i}X_{i}$. Then $(\xi_{i})_{i\in I}$ are independent and

From Example 2.4, the variables $(y_{i})_{i\in I}$ defined by

are half-independent, and $y_{i}$ has the same distribution as $x_{i}$ for each $i\in I$. By Lemma 2.7, $(y_{i})_{i\in I}$ has the same joint distribution as $(x_{i})_{i\in I}$.

We will now develop a combinatorial theory for half-independence, based on the family $E_{h}$ of balanced partitions.

Let $(\mathcal{A},E\dvtx\mathcal{A}\to\mathcal{B})$ be an operator-valued probability space, and suppose that $\mathcal{B}$ is contained in the center of $\mathcal{A}$. Let $(x_{i})_{i\in I}$ be a family of random-variables in $\mathcal{A}$, and suppose that

for any odd $k$ and $i_{1},\ldots,i_{k}\in I$. Define the half-liberated cumulants $\gamma_{E}^{(n)}$ by the half-liberated moment-cumulant formula

where $\gamma_{E}^{(\pi)}[x_{i_{1}},\ldots,x_{i_{k}}]$ is defined, as in the classical case, by the formula

Observe that both sides of the moment-cumulant formula above are equal to zero for odd values of $k$, and for even values the right hand side is equal to $\gamma_{E}^{(k)}[x_{i_{1}},\ldots,x_{i_{k}}]$ plus products of lower ordered terms and hence $\gamma_{E}^{(k)}$ may be solved for recursively. As in the free and classical cases, we may apply the Möbius inversion formula to obtain the following equation for $\gamma_{E}^{(\pi)}$, $\pi\in E_{h}(k)$:

$E[x_{i_{1}}\cdots x_{i_{k}}]=0$ whenever $k$ is odd, and

for any $\pi\in E_{h}(k)$ such that $\pi\not\leq\ker\mathbf{i}$.

First, suppose that condition (2) holds. From the moment-cumulant formula, we have

so that $(x_{i}^{2})_{i\in I}$ are independent and hence $(x_{i})_{i\in I}$ are half-independent.

The implication $(1)\Rightarrow(2)$ actually follows from $(2)\Rightarrow(1)$. Indeed, suppose that $(x_{i})_{i\in I}$ are half-independent. Consider the algebra $\mathcal{A}^{\prime}=\mathcal{B}\langle y_{i}\dvtx i\in I\rangle/\langle y_{i}y_{j}y_{k}=y_{k}y_{j}y_{i}\rangle$ of polynomials in half-commuting indeterminates $(y_{i})_{i\in I}$ and coefficients in $\mathcal{B}$. Define a conditional expectation $E^{\prime}\dvtx\mathcal{A}^{\prime}\to\mathcal{B}$ by

(It is easy to see that $E^{\prime}$ is well defined, that is, compatible with the half-commutation relations.) Since the half-liberated cumulants are uniquely determined by the moment-cumulant formula, it follows that

By the first part, it follows that $(y_{i})_{i\in I}$ are half-independent with respect to $E^{\prime}$. Since $y_{i}$ has the same $\mathcal{B}$-valued distribution as $x_{i}$, it follows from Lemma 2.7 that $(y_{i})_{i\in I}$ have the same joint distribution as $(x_{i})_{i\in I}$. It then follows from the moment-cumulant formula that these families have the same half-liberated cumulants, and hence $\gamma_{E}^{(\pi)}[x_{i_{1}},\ldots,x_{i_{k}}]=0$ unless $\pi\leq\ker\mathbf{i}$.

Recall that (centered) Gaussian and semicircular distributions are characterized by the property that their nonvanishing cumulants are those corresponding to pair and noncrossing pair partitions, respectively. We will now show that for half-independence, it is the symmetrized Rayleigh distribution which has this property. This follows from the considerations in bsp , but we include here a direct proof.

Let $x$ be a random variable in $(\mathcal{A},\varphi)$ which has an even distribution. Then $x$ has a symmetrized Rayleigh distribution if and only if

for any $\pi\in E_{h}(k)$ such that $\pi\notin E_{2}(k)$.

Since the distribution of $x$ is determined uniquely by its half-liberated cumulants, it suffices to show that if the cumulants have the stated property then $x$ has a symmetrized Rayleigh distribution. Suppose that this is the case, then

It is easy to see that the number of partitions in $E_{2}(k)$ is $m!$ if $k=2m$ is even and is zero if $k$ is odd. Since these agree with the moments of a symmetrized Rayleigh distribution, the result follows.

Weingarten estimate

If $G\subset O_{n}$ is a compact group, then the Haar state on $C(G)$ is given by integrating against the Haar measure on $G$.

One quite useful aspect of the easiness condition for a compact orthogonal quantum group is that it leads to a combinatorial Weingarten formula for computing the Haar state, which we now recall from bsp .

$G_{kn}$ is invertible for $n$ sufficiently large (see Proposition 3.4), define the Weingarten matrix $W_{kn}$ to be its inverse.

Let $G\subset O_{n}^{+}$ be an easy quantum group and let $D(k)\subset P(0,k)$ be the corresponding collection of partitions having no upper points. If $G_{kn}$ is invertible, then

The statement of the theorem above is from bsp , but goes back to work of Weingarten wein and was developed in a series of papers col , cos , bc1 , bc2 . Note that this reduces the problem of evaluating integrals over $G$ to computing the entries of the Weingarten matrix. We will now give an estimate on the asymptotic behavior of $W_{kn}$ as $n\to\infty$. This unifies and extends the estimates given in bc1 and cur3 for $O^{+},S^{+}$.

$W_{kn}(\pi,\sigma)=O(n^{|\pi\vee\sigma|-|\pi|-|\sigma|})$.

where $\mu_{D(k)}$ is the Möbius function on the partially ordered set $D(k)$ under the restriction of the order on $P(k)$.

We use a standard method from col , cos , further developed in bc1 , bc2 , cur1 .

Note that the entries of $B_{kn}$ are $O(n^{-1/2})$, it follows that for $n$ sufficiently large $1+B_{kn}$ is invertible and

So to prove (1), it suffices to show that if $\nu_{1},\ldots,\nu_{l}\in D(k)$, then

We will use the fact that $P(k)$ is a semi-modular lattice (birk , Section I.8, Example 9): If $\nu,\tau\in\mathcal{P}(k)$, then

We will now prove the claim by induction on $l$, for $l=1$ we may apply the formula above to find

Also $|\nu_{l}\vee\sigma|\leq|\pi\vee\sigma|+|\nu_{l}|-|\pi\vee\nu_{l}|$, and the result follows.

To prove (2), suppose $\pi,\sigma\in D(k)$ and $\pi\leq\sigma$. The terms which contribute to order $n^{-|\pi|}$ in the expansion come from sequences $\nu_{1},\ldots,\nu_{l}\in D(k)$ such that $\pi\neq\nu_{1}\neq\cdots\neq\nu_{l}\neq\sigma$ and

Since $|\pi\vee\nu_{1}|\leq|\nu_{1}|$, $|\nu_{1}\vee\nu_{2}|\leq|\nu_{2}|,\ldots,|\nu_{l}\vee\sigma|\leq\sigma$, it follows that each of these must be an equality, which implies $\pi<\nu_{1}<\cdots<\nu_{l}<\sigma$. Conversely, any $\nu_{1},\ldots,\nu_{l}\in D(k)$ such that $\pi<\nu_{1}<\cdots<\nu_{l}<\sigma$ clearly satisfy this equation. Therefore, the coefficient of $n^{-|\pi|}$ in $W_{kn}(\pi,\sigma)$ is

which is precisely $\mu_{D(k)}(\pi,\sigma)$.

Recall that the free, half-liberated and classical cumulants are obtained from moment functionals by using the Möbius functions on $\mathit{NC},E_{h}$ and $P$, respectively. To show that this is compatible with Proposition 3.4, we will need the following result.

If $D=\mathit{NC},\mathit{NC}_{2},\mathit{NC}_{b},\mathit{NC}_{h}$, then

Let $Q=\mathit{NC},E_{h},P$ according to cases (1), (2), (3). It is easy to see in each case that $D(k)$ is closed under taking intervals in $Q(k)$, that is, if $\pi_{1},\pi_{2}\in D(k)$, $\sigma\in Q(k)$ and $\pi_{1}<\sigma<\pi_{2}$ then $\sigma\in D(k)$. The result now follows immediately from the definition of the Möbius function.

Finite quantum invariant sequences

We begin this section by defining the notion of quantum invariance for a sequence of noncommutative random variables under “transformations” coming from an orthogonal quantum group $G_{n}\subset O_{n}^{+}$.

It is easily verified that $\alpha_{n}$ is an action of $G_{n}$, that is,

for all $p\in\mathscr{P}_{n}$. More explicitly, the sequence $(x_{1},\ldots,x_{n})$ is $G_{n}$-invariant if

Suppose that $G_{n}\subset O_{n}$ is a compact group. By evaluating both sides of the above equation at $g\in G_{n}$, we see that a sequence $(x_{1},\ldots,x_{n})$ is $G_{n}$-invariant if and only if

We will now prove a converse to Theorem 1, which holds for finite sequences and in a purely algebraic context. The proof is adapted from the method of ksp , Proposition 3.1.

Let $(\mathcal{A},\varphi)$ be a noncommutative probability space, $1\in\mathcal{B}\subset\mathcal{A}$ a unital subalgebra and $E\dvtx\mathcal{A}\to\mathcal{B}$ a conditional expectation which preserves $\varphi$. Let $(x_{1},\ldots,x_{n})$ be a sequence in $\mathcal{A}$.

If $x_{1},\ldots,x_{n}$ are freely independent and identically distributed with amalgamation over $\mathcal{B}$, then the sequence is $S_{n}^{+}$-invariant.

If $x_{1},\ldots,x_{n}$ are freely independent and identically distributed with amalgamation over $\mathcal{B}$, and have even distributions with respect to $E$, then the sequence is $H_{n}^{+}$-invariant.

If $x_{1},\ldots,x_{n}$ are freely independent and identically distributed with amalgamation over $\mathcal{B}$, and have semicircular distributions with respect to $E$, then the sequence is $B_{n}^{+}$-invariant.

If $x_{1},\ldots,x_{n}$ are freely independent and identically distributed with amalgamation over $\mathcal{B}$, and have centered semicircular distributions with respect to $E$, then the sequence is $O_{n}^{+}$-invariant.

Half-liberated case: Suppose that $(x_{1},\ldots,x_{n})$ half-commute, and that $\mathcal{B}$ is central in $\mathcal{A}$. {longlist}[(b)]

If $x_{1},\ldots,x_{n}$ are half-independent and identically distributedgiven $\mathcal{B}$, then the sequence is $H_{n}^{*}$-invariant.

If $x_{1},\ldots,x_{n}$ are half-independent and identically distributedgiven $\mathcal{B}$, and have symmetrized Rayleigh distributions with respect to $E$, then the sequence is $O_{n}^{*}$-invariant.

Suppose that $\mathcal{B}$ and $x_{1},\ldots,x_{n}$ generate a commutative algebra. {longlist}[(b)]

If $x_{1},\ldots,x_{n}$ are conditionally independent and identically distributed given $\mathcal{B}$, then the sequence is $S_{n}$-invariant.

If $x_{1},\ldots,x_{n}$ are conditionally independent and identically distributed given $\mathcal{B}$, and have even distributions with respect to $E$, then the sequence is $H_{n}$-invariant.

If $x_{1},\ldots,x_{n}$ are conditionally independent and identically distributed given $\mathcal{B}$, and have Gaussian distributions with respect to $E$, then the sequence is $B_{n}$-invariant.

If $x_{1},\ldots,x_{n}$ are conditionally independent and identically distributed given $\mathcal{B}$, and have centered Gaussian distributions with respect to $E$, then the sequence is $O_{n}$-invariant.

where $\xi$ denotes the free, half-liberated or classical cumulants in cases (1), (2) and (3), respectively. It follows from the considerations in bsp , or by direct computation, that if $\pi\in D(k)$ then

To prove the approximation result for finite sequences, we will require more analytic structure. Throughout the rest of the section, we will assume that $G_{n}\subset O_{n}^{+}$ is a compact quantum group, $(M,\varphi)$ is a W∗-probability space and $(x_{1},\ldots,x_{n})$ is a sequence of self-adjoint random variables in $M$. We denote the von Neumann algebra generated by $(x_{1},\ldots,x_{n})$ by $M_{n}$, and define the $G_{n}$-invariant subalgebra by

where $\mathscr{P}_{n}^{\alpha_{n}}$ denotes the fixed point algebra of the action $\alpha_{n}$, that is,

We now begin the technical preparations for our approximation result. First, we will need to extend the action $\alpha_{n}$ to the von Neumann algebra context. $L^{\infty}(G_{n})$ will denote the von Neumann algebra obtained by taking the weak closure of $\pi_{n}(C(G_{n}))$, where $\pi_{n}$ is the GNS representation of $C(G_{n})$ on the GNS Hilbert space $L^{2}(G_{n})$ for the Haar state. $L^{\infty}(G_{n})$ is a Hopf von Neumann algebra, with the natural structure induced from $C(G_{n})$.

Suppose that $(x_{1},\ldots,x_{n})$ is $G_{n}$-invariant. Then there is a right coaction $\widetilde{\alpha}_{n}\dvtx M_{n}\to M_{n}\otimes L^{\infty}(G_{n})$ determined by

for $p\in\mathscr{P}_{n}$. Moreover, the fixed point algebra of $\widetilde{\alpha}_{n}$ is precisely the $G_{n}$-invariant subalgebra $\mathcal{B}_{n}$.

There is a natural conditional expectation $E_{n}\dvtx M_{n}\to\mathcal{B}_{n}$ given by integrating the coaction $\widetilde{\alpha}_{n}$ with respect to the Haar state, that is,

By using the Weingarten calculus, we can give a simple combinatorial formula for the moment functionals with respect to $E_{n}$ if $G_{n}$ is one of the easy quantum groups under consideration. In the half-liberated case, we must first show that $\mathcal{B}_{n}$ is central in $M_{n}$.

Suppose that $(x_{1},\ldots,x_{n})$ half-commute. If $H_{n}^{*}\subset G_{n}$, then the $G_{n}$-invariant subalgebra $\mathcal{B}_{n}$ is contained in the center of $M_{n}$.

Since $x_{i}^{2}$ is central in $M_{n}$ for $1\leq i\leq n$, the result follows.

Suppose that $(x_{1},\ldots,x_{n})$ is $G_{n}$-invariant, and that one of the following conditions is satisfied: {longlist}[(3)]

$G_{n}$ is a free quantum group $O_{n}^{+},S_{n}^{+},H_{n}^{+}$ or $B_{n}^{+}$.

$G_{n}$ is a half-liberated quantum group $O_{n}^{*}$ or $H_{n}^{*}$ and $(x_{1},\ldots,x_{n})$ half-commute.

$G_{n}$ is an easy group $O_{n},S_{n},H_{n}$ or $B_{n}$ and $(x_{1},\ldots,x_{n})$ commute. Then for any $\pi$ in the partition category $D(k)$ for the easy quantum group $G_{n}$, and any $b_{0},\ldots,b_{k}\in\mathcal{B}_{n}$, we have

which holds if $n$ is sufficiently large that the Gram matrix $G_{kn}$ is invertible.

We prove this by induction on the number of blocks of $\pi$. First, suppose that $\pi=1_{k}$ is the partition with only one block. Then

where we have used the fact that $b_{0},\ldots,b_{k}$ are fixed by the coaction $\widetilde{\alpha}_{n}$. Applying the Weingarten integration formula in Proposition 3.2, we have

Observe that $G_{kn}(\sigma,1_{k})=n^{|\sigma\vee 1_{k}|}=n$ for any $\sigma\in D(k)$. It follows that for any $\pi\in D(k)$, we have

If condition (2) or (3) are satisfied, then the general case follows from the formula

where in the half-liberated case we are applying the previous lemma. The one thing we must check here is that if $\pi\in D(k)$ and $V$ is a block of $\pi$ with $s$ elements, then $1_{s}\in D(s)$. This is easily verified, in each case, for $D=P,P_{2},P_{h},P_{b},E_{h},E_{2}$.

Suppose now that condition (1) is satisfied. Let $\pi\in D(k)$. Since $\pi$ is noncrossing, $\pi$ contains an interval $V=\{l+1,\ldots,l+s+1\}$. We then have

To apply induction, we must check that $\pi\setminus V\in D(k-s)$ and $1_{s}\in D(s)$. Indeed, this is easily verified for $\mathit{NC},\mathit{NC}_{2},\mathit{NC}_{h}$ and $\mathit{NC}_{b}$. Applying induction, we have

We are now prepared to prove the approximation result for finite sequences.

Suppose that $(x_{1},\ldots,x_{n})$ is $G_{n}$-invariant, and that one of the following conditions is satisfied: {longlist}[(3)]

$G_{n}$ is a free quantum group $O_{n}^{+},S_{n}^{+},H_{n}^{+}$ or $B_{n}^{+}$.

$G_{n}$ is a half-liberated quantum group $O_{n}^{*}$ or $H_{n}^{*}$ and $(x_{1},\ldots,x_{n})$ half-commute.

$G_{n}$ is an easy group $O_{n},S_{n},H_{n}$ or $B_{n}$ and $(x_{1},\ldots,x_{n})$ commute. Let $(y_{1},\ldots,y_{n})$ be a sequence of $\mathcal{B}_{n}$-valued random variables with $\mathcal{B}_{n}$-valued joint distribution determined as follows:

$G=O^{+}$: Free semicircular, centered with same variance as $x_{1}$.

$G=S^{+}$: Freely independent, $y_{i}$ has same distribution as $x_{1}$.

$G=H^{+}$: Freely independent, $y_{i}$ has same distribution as $x_{1}$.

$G=B^{+}$: Free semicircular, same mean and variance as $x_{1}$.

$G=O^{*}$: Half-liberated Gaussian, same variance as $x_{1}$.

$G=H^{*}$: Half-independent, $y_{i}$ has same distribution as $x_{1}$.

$G=O$: Independent Gaussian, centered with same variance as $x_{1}$.

$G=S$: Independent, $y_{i}$ has same distribution as $x_{1}$.

$G=H$: Independent, $y_{i}$ has same distribution as $x_{1}$.

$G=B$: Independent Gaussian, same mean and variance as $x_{1}$.

If $1\leq j_{1},\ldots,j_{k}\leq n$ and $b_{0},\ldots,b_{k}\in\mathcal{B}_{n}$, then

where $C_{k}(G)$ is a universal constant which depends only on $k$ and the easy quantum group $G$.

First, we note that it suffices to prove the statement for $n$ sufficiently large, in particular we will assume throughout that $n$ is sufficiently large for the Gram matrix $G_{kn}$ to be invertible.

Let $1\leq j_{1},\ldots,j_{k}\leq n$ and $b_{0},\ldots,b_{k}\in\mathcal{B}_{n}$. We have

On the other hand, it follows from the assumptions on $(y_{1},\ldots,y_{n})$ and the various moment-cumulant formulae that

where $\xi$ denotes the relevant free, classical or half-liberated cumulants. The right-hand side can be expanded, via Möbius inversion, in terms of expectation functionals $E_{n}^{(\pi)}[b_{0}x_{1}b_{1},\ldots,x_{1}b_{k}]$ where $\pi$ is a partition in $\mathit{NC},E_{h},P$ according to cases (1), (2), (3), and $\pi\leq\sigma$ for some $\sigma\in D(k)$. Now if $\pi\notin D(k)$ then we claim that this expectation functional is zero. Indeed this is only possible if $D=\mathit{NC}_{2},\mathit{NC}_{h},P_{2},P_{h}$ and $\pi$ has a block with an odd number of legs. But it is easy to see that in these cases $x_{1}$ has an even distribution with respect to $E_{n}$, and therefore $E_{n}^{(\pi)}[b_{0}x_{1}b_{1},\ldots,x_{1}b_{k}]=0$ as claimed. This observation, together with Proposition 3.5, allows to to rewrite the above equation as

Comparing these two equations, we find that

Now since $x_{1},\ldots,x_{n}$ are identically distributed with respect to the faithful state $\varphi$, it follows that these variables have the same norm. Therefore,

for any $\pi\in D(k)$. Combining this with former equation, we have

which is finite by Proposition 3.4, completes the proof.

Infinite quantum invariant sequences

In this section, we will prove Theorem 1. Throughout this section, we will assume that $G$ is one of the easy quantum groups $O,S,H,B,O^{*},H^{*},O^{+},S^{+},H^{+}$ or $B^{+}$. We will make use of the inclusions $G_{n}\hookrightarrow G_{m}$ for $n<m$, which correspond to the Hopf algebra morphisms $\omega_{n,m}\dvtx C(G_{m})\to C(G_{n})$ determined by

The existence of $\omega_{n,m}$ may be verified in each case by using the universal relations of $C(G_{n})$.

We begin by extending the notion of $G_{n}$-invariance to infinite sequences.

It is clear that $\beta_{n}$ is an action of $G_{n}$, moreover we have the relations

where $\mathscr{P}_{\infty}^{\beta_{n}}$ is the fixed point algebra of the action $\beta_{n}$. Since

it follows that $\mathcal{B}_{n+1}\subset\mathcal{B}_{n}$ for all $n\geq 1$. We then define the $G$-invariant subalgebra by

for $m\in M$. By taking the limit as $n\to\infty$, we obtain a $\varphi$-preserving conditional expectation onto the $G$-invariant subalgebra.