Hopf algebras and the logarithm of the S-transform in free probability

0.41 MB·English

by  Mitja Mastnak

Hopf algebras and the logarithm of the S-transform
in free probability
Mitja Mastnak Alexandru Nica 1
9
0
Abstract
0
2
Letk be apositiveintegerandletG denotethe setofalljointdistributions ofk-tuples
k

n (a ,...,a ) in a non-commutative probability space (A,ϕ) such that ϕ(a ) = ··· =
1 k 1
a ϕ(a )=1. G is a group under the operation of free multiplicative convolution ⊠. We
k k
J identify Gk,⊠ as the group of characters of a certain Hopf algebra Y(k). Then, by
9
using the(cid:0) log m(cid:1)ap from characters to infinitesimal characters of Y(k), we introduce a
2
transform LSµ for distributions µ ∈ Gk. LSµ is a power series in k non-commuting

] indeterminates z ,...,z ; its coefficients can be computed from the coefficients of the
1 k
A
R-transformofµbyusingsummationsoverchainsinthelatticesNC(n)ofnon-crossing
O partitions. The LS-transformhas the “linearizing” property that
.
th LSµ⊠ν =LSµ+LSν, ∀µ,ν ∈Gk such that µ⊠ν =ν⊠µ.
a
m In the particular case k = 1 one has that Y(1) is naturally isomorphic to the Hopf
[ algebra Sym of symmetric functions, and that the LS-transformis very closely related

to the logarithm of the S-transform of Voiculescu, by the formula
2
v
LS (z)=−zlogS (z), ∀µ∈G .
9 µ µ 1
6
1 In this case the group (G1,⊠) can be identified as the group of characters of Sym, in
4 such a way that the S-transform, its reciprocal 1/S and its logarithm logS relate in
7. a natural sense to the sequences of complete, elementary and respectively power sum
0 symmetric functions.
8
0
:
v 1. Introduction
i
X
r In this paper we study joint distributions of k-tuples of elements in a noncommutative
a
probability space. Let (A,ϕ) be a noncommutative probability space (i.e. A is a unital
algebra over C and ϕ :A → C is a linear functional such that ϕ(1 ) = 1), and let a ,...,a
A 1 k
be elements of A. The distribution of (a ,...,a ) is the linear functional µ on the algebra
1 k
of noncommutative polynomials ChX ,...,X i defined by the requirement that
1 k
µ(X ···X )= ϕ(a ···a ), ∀n ≥ 0, ∀1≤ i ,...,i ≤k.
i1 in i1 in 1 n
We denote by D (k) the set of linear functionals on ChX ,...,X i that arise in this way.
alg 1 k
(Clearly, this is just the set of all linear functionals on ChX ,...,X i such that µ(1) = 1.)
1 k
On D (k) we have a binary operation ⊠ which reflects the multiplication of two freely
alg
independent k-tuples in a noncommutative probability space. That is, ⊠ is well-defined
and uniquely determined by the following requirement: if a ,...,a ,b ,...,b are elements
1 k 1 k
in a noncommutative probability space (A,ϕ) such that (a ,...,a ) has distribution µ,
1 k
(b ,...,b ) has distribution ν, and {a ,...,a } is freely independent from {b ,...,b },
1 k 1 k 1 k
1Research supported by a Discovery Grant from NSERC, Canada.
1
then it follows that the distribution of (a b ,...,a b ) is equal to µ ⊠ ν. The operation
1 1 k k
⊠ on D (k) is associative and unital, where the unit is the functional µ ∈ D (k) with
alg o alg
µ (X ···X ) = 1 for all n ≥ 1 and 1 ≤ i ,...,i ≤ k. A distribution µ ∈ D (k) is
o i1 in 1 n alg
invertible with respect to ⊠ if and only if it satisfies µ(X ) 6= 0, ∀1≤ i≤ k; and moreover,
i
the subset
G := {µ ∈ D (k) | µ(X ) = 1, ∀1 ≤ i≤ k} (1.1)
k alg i
is a subgroup in the group of invertibles with respect to ⊠. For a basic introduction to free
multiplicative convolution, we refer to Section 3.6 of [17] or to Lecture 14 in [11].
Themain goal of thepresentpaperis tointroduceatransformLS fordistributionsµ ∈
µ
G , which linearizes commuting ⊠-products. We arrive to the LS-transform by identifying
k
G ,⊠ as the group of characters of a certain Hopf algebra Y(k). As an algebra, Y(k) is
k
(cid:0)merely(cid:1)a commutative algebra of polynomials:
Y(k) = C Y |w ∈ [k]∗, |w| ≥ 2 , (1.2)
w
(cid:2) (cid:3)
where [k]∗ is the set of all words of finite length made with letters from the alphabet
{1,...,k}, and where |w| denotes the length (i.e. number of letters) of a word w ∈ [k]∗.
The feature which relates Y(k) to free probability is its comultiplication: for w ∈ [k]∗ with
|w| = n ≥ 2, the comultiplication ∆(Y ) ∈ Y(k) ⊗Y(k) is defined by using a special type
w
of summation over the lattice of non-crossing partitions NC(n) which is known (see e.g.
Lectures 14 and 17 of [11]) to relate to the multiplication of free random variables. The
details of the construction of Y(k) are presented in Section 3 below. Here we only mention
the fact that Y(k) is a graded connected Hopf algebra, where every generator Y from (1.2)
w
is homogeneous of degree |w|−1. A consequence of this fact which is important for the
present paper is that one can then define the exponential expξ for every linear functional
ξ : Y(k) → C such that ξ(1) = 0, and one can define the logarithm logη for every linear
functional η : Y(k) → C such that η(1) = 1. Moreover, the maps exp and log are bijections
inverse to each other, between the two sets of linear functionals mentioned above.
Let X(Y(k)) be the set of characters of Y(k),
X(Y(k)):= η :Y(k) → C | η(1) = 1, η is linear and multiplicative . (1.3)
n o
Then the logarithm map sends X(Y(k)) bijectively onto the set I(Y(k)) of infinitesimal char-
acters of Y(k); the latter set is defined by
ξ is linear and satisfies
I(Y(k)):= ξ :Y(k) → C , (1.4)
(cid:26) ξ(PQ) = ξ(P)ε(Q)+ε(P)ξ(Q), ∀P,Q ∈ Y(k) (cid:27)
where ε is the counit of Y(k) (ε ∈ X(Y(k)), and is uniquely determined by the requirement
that ε(Y ) = 0 for every w ∈ [k]∗ with |w| ≥ 2). Let us also record the facts that X(Y(k)) is
w
a group under the operation of convolution for linear functionals on Y(k), and that the log
map linearizes commuting convolution products:
η ,η ∈ X(Y(k)), η η = η η ⇒ log(η η ) = logη +logη . (1.5)
1 2 1 2 2 1 1 2 1 2
(cid:16) (cid:17)
(In this paper the convolution of linear functionals on a Hopf algebra is denoted as a plain
multiplication. Our conventions of notation and a few background facts about graded
connected Hopf algebras are collected in Section 2C of the paper.)
2
So now let us state precisely what is the connection between the Hopf algebra Y(k) and
the operation ⊠. Let µ be a distribution in G , and let us consider its R-transform R ; this
k µ
is a power series in k non-commuting indeterminates z ,...,z , of the form
1 k
k
R (z ,...,z ) = z + α z , (1.6)
µ 1 k i w w
Xi=1 wX∈[k]∗,
|w|≥2
whereinthelattersumweusedtheshorthandnotationz := z ···z forw = (i ,...,i ) ∈
w i1 in 1 n
[k]∗ with n ≥ 2. A brief review of how the coefficients α are calculated from the moments
w
(i.e. values on monomials) of µ is made in Section 2B below – see Definition 2.5, Remark
2.6.
Definition 1.1. Let µ be a distribution in G and consider the R-transform R , denoted
k µ
as in Equation (1.6). The character of Y(k) associated to µ is the character χ ∈ X(Y(k))
µ
uniquely determined by the requirement that
χ (Y )= α , ∀w ∈ [k]∗ such that |w| ≥ 2. (1.7)
µ w w
Theorem 1.2. The map µ 7→ χ defined above is a group isomorphism from G ,⊠ onto
µ k
the group X(Y(k)) of characters of Y(k) (endowed with the operation of convol(cid:0)ution).(cid:1)
The proof of Theorem 1.2 is presented in Section 3 below. Let us mention that the
proof is quite short, and follows easily from known facts about how the multiplication of
freek-tuplesisdescribedintermsoftheirR-transforms,asexplainedforinstanceinLecture
17 of [11]. (In a certain sense, we are dealing here with a situation where the convolution
of characters of Y(k) was studied before looking at Y(k) itself.) Nevertheless, Theorem 1.2
is important because it brings to attention the fact that Hopf algebra methods can be used
in the study of ⊠. In particular, by taking (1.5) into account, we see that Theorem 1.2 has
the following immediate corollary.
Corollary 1.3. Let µ and ν be distributions in G such that µ⊠ν = ν ⊠µ. Then
k
logχµ⊠ν = logχµ+logχν, (1.8)
where the characters χµ, χν and χµ⊠ν are as in Definition 1.1, and their logarithms are the
corresponding functionals from I(Y(k)).
TheLS-transformLS isdefinedsothatitstorestheinformationabouttheinfinitesimal
µ
character logχ , as follows.
µ
Definition 1.4. Let µ be a distribution in G . The LS-transform of µ is the power series
k
LS (z ,...,z ) := logχ (Y ) z , (1.9)
µ 1 k µ w w
wX∈[k]∗, (cid:16)(cid:0) (cid:1) (cid:17)
|w|≥2
where logχ ∈ I(Y(k)) is as in Corollary 1.3, and where the meaning of “z ” is same as in
µ w
Equation (1.6).
Clearly, Corollary 1.3 can also be phrased as a statement about LS-transforms. In this
guise, it simply says that the LS-transform linearizes commuting ⊠-products:
3
Corollary 1.5. Let µ and ν be distributions in G such that µ⊠ν = ν ⊠µ. Then
k
LSµ⊠ν = LSµ+LSν. (1.10)
Inparticular, formula(1.10)always applieswhenoneofµ,ν is thejointdistributionof a
repeatedk-tuple(a,a,...,a),whereaisarandomvariableinanon-commutativeprobability
space (A,ϕ). A version of this fact which lives in the framework of a C∗-probability space
is discussed in Example 5.2 below.
The LS-transform was introduced above by using the Hopf algebra Y(k), but it can also
be described directly in combinatorial terms, by using summations over chains in lattices
of non-crossing partitions. We next explain how this goes.
A chain in NC(n) is an object of the form
Γ = (π ,π ,...,π ) (1.11)
0 1 ℓ
with π ,π ,...,π ∈ NC(n) such that 0 = π < π < ··· < π = 1 (and where 0
0 1 ℓ n 0 1 ℓ n n
and 1 denote the minimal and maximal element of NC(n), respectively). For a chain Γ
n
as in (1.11), the number ℓ is called the length of Γ and will be denoted as |Γ|. Given a
formal power series f in non-commuting indeterminates z ,...,z , one has a natural way
1 k
of defining some “generalized coefficients”
(Γ)
Cf (f) (1.12)
(i1,...,in)
where n ≥ 2, 1 ≤ i ,...,i ≤ k, and Γ is a chain in NC(n). Every generalized coefficient
1 n
(1.12) is defined to be a certain product of actual coefficients of f (see Definition 4.3 below
for the precise formula). By using the generalized coefficients (1.12), the combinatorial
description of the LS-transform is stated as follows.
Theorem 1.6. Let µ be a distribution in G , and let w = (i ,...,i ) be a word in [k]∗,
k 1 n
where n ≥ 2. Consider (as in Equation (1.9) of Definition 1.4) the coefficient logχ (Y )
µ w
of z in the LS-transform of µ. This coefficient can be also expressed as (cid:0) (cid:1)
w
(−1)1+|Γ|
(Γ)
logχ (Y ) = Cf (R ), (1.13)
µ w |Γ| (i1,...,in) µ
(cid:0) (cid:1) X
Γ chain
in NC(n)
where R is the R-transform of µ.
µ
Remark 1.7. 1o NC(n) has a unique chain of length 1 (namely the chain (0 ,1 )); the
n n
term indexed by this chain in the sum on the right-hand side of (1.13) is precisely the
coefficient of z in R , while every other term of the same sum turns out to bea product of
w µ
coefficients ofR withlengthsstrictly smaller thann. Fromthisobservation itisimmediate
µ
that the coefficients of R can be computed back, recursively, in terms of the coefficients of
µ
LS . Moreover, since µ is completely determined by its R-transform, we thus see that µ is
µ
completely determined by the series LS as well.
µ
2o By using Theorem 1.6 one finds that the LS-transform inherits a fundamental prop-
erty that the R-transform is known to have in connection to free independence: a k-tuple in
a noncommutative probability space is freely independent if and only if the LS-transform
of its distribution separates the variables. The precise statement of this fact appears as
Proposition 5.4 below.
4
In the remaining part of the introduction we look at the particular case when k = 1. In
this case the notations are simplified due to the fact that words over the 1-letter alphabet
{1} are determined by their lengths. We make the convention to write simply “Y ” instead
n
of Y with n repetitions of 1 in the index; thus Equation (1.2) is now written in the
(1,1,...,1)
form
Y(1) = C Y |n ≥ 2 , (1.14)
n
(cid:2) (cid:3)
while Equation (1.9) defining LS reduces to
µ
∞
LS (z) = (logχ )(Y ) zn. (1.15)
µ µ n
nX=2(cid:16) (cid:17)
A special feature of the case k = 1 (not holding for k ≥ 2) is that the operation ⊠ is
commutative. Hence for k = 1 the linearization property stated in Corollary 1.5 holds for
all µ,ν ∈ G .
1
Now, in the case k = 1 there exists an established way of treating multiplicative free
convolution,byusingVoiculescu’sS-transform. TheS-transformS ofadistributionµ ∈G
µ 1
is a power series in an indeterminate z, with constant term equal to 1. Taking S-transforms
converts ⊠ into plain multiplication of power series:
Sµ⊠ν(z) = Sµ(z)·Sν(z), ∀µ,ν ∈ G1 (1.16)
(see Section 3.6 in [17], or Lecture 18 in [11]). We prove that the one-dimensional LS-
transform is related to the S-transform, as follows.
Theorem 1.8. For a distribution µ ∈G , the power series S and LS are related by
1 µ µ
LS (z) = −zlogS (z). (1.17)
µ µ
Theorem1.8showsthatthemultiplicativity oftheS-transformcanberetrievedfromthe
particularcasek = 1ofCorollary1.5: oneonlyneedstodivideby−z andthenexponentiate
both sides of Equation (1.10). We should note here that this exponentiating trick is specific
tothe1-variablesituation,fork ≥ 2itisprobablybettertoworkwithLS itselfratherthan
µ
considering its exponential. (Indeed, for k ≥ 2 it isn’t generally true that the k-variable
series LS and LS would commute, even if µ,ν ∈ G are such that µ⊠ν = ν ⊠µ. Hence
µ ν k
when one exponentiates Equation (1.10) for k ≥ 2, the series appearing on the right-hand
side isn’t generally equal to exp LS ·exp LS .)
µ ν
The proof of Theorem 1.8 is(cid:0)obta(cid:1)ined b(cid:0)y fol(cid:1)lowing the connections that Y(1) has with
symmetric functions. We start from the fact that (independently of the isomorphism G ∋
1
µ 7→ χ ∈ X(Y(1)) from Theorem 1.2) one can also find a natural group isomorphism
µ
µ 7→ θ from (G ,⊠) onto the group of characters of the Hopf algebra Sym of symmetric
µ 1
functions. Themapµ 7→ θ isdefinedinsuchawaythattheS-transform,itsreciprocal1/S
µ
and its logarithm logS relate in a natural sense to the sequences of complete, elementary
and respectively power sum symmetric functions (see Remark 6.7 below for the precise
description of how this happens). We then concretely put into evidence an isomorphism
Φ :Y(1) → Sym which has the property that
χ = θ ◦Φ, ∀µ ∈ G ; (1.18)
µ µ 1
with the help of Φ we can place the 1-dimensional LS-transform in the framework of Sym,
and then prove the relation stated in Theorem 1.8 (this is done in Proposition 6.10 and
Corollary 6.12).
5
ItisinterestingtonotethatthewaytheLS-transformisintroducedinthispaperisclose
inspirittohowtheS-transformwasfirstfoundbyVoiculescuin[16]. Someotherapproaches
to theS-transform werefoundin themeanwhile(e.g. in[6]or in [10]), butthe onefrom [16]
has the distinctive feature that it relies on the exponential map for a certain commutative
Lie group, constructed to reflect how the moments of µ ⊠ ν are computed in terms of
the moments of µ and of ν (µ,ν ∈ G ). The approach in the present paper goes on the
1
same lines, with the difference that it insists on structures (free cumulants, Hopf algebras)
where it is easier to pursue a detailed combinatorial analysis. The possibility of moving up
to LS-transforms for k-tuples then comes as a manifestation of the general principle that
combinatorial arguments in free probability often extend without much trouble from the
1-variable to the k-variable setting.
We conclude this introduction by describing how the paper is organized. Besides the in-
troduction,thepaperhassixothersections. Section2containsareviewofsomebackground
and notations. In Section 3 we introduce the Hopf algebra Y(k) and we prove Theorem 1.2,
then Section 4 is devoted to the combinatorics of the LS-transform and to the proof of
Theorem 1.6. In Section 5 we discuss some basic properties of LS . The last two sections
µ
of the paper are devoted to the one-variable framework: in Section 6 we prove Theorem
1.8 and in Section 7 we discuss in more detail the isomorphism between Y(1) and the Hopf
algebra Sym of symmetric functions.
2. Background and notations
2A. Non-crossing partitions
Notation 2.1. 1o We will use the standard conventions of notation for non-crossing parti-
tions (as in [13], or in Lecture 9 of [11]). For a positive integer n, the set of all non-crossing
partitions of {1,...,n} will be denoted by NC(n). For π ∈ NC(n), the number of blocks
of π will be denoted by |π|. On NC(n) we consider the partial order given by reversed
refinement: for π,ρ ∈ NC(n), we write “π ≤ ρ” to mean that every block of ρ is a union
of blocks of π. The minimal and maximal element of (NC(n),≤) are denoted by 0 (the
n
partition of {1,...,n} into n blocks of 1 element each) and respectively 1 (the partition
n
of {1,...,n} into 1 block of n elements).
2o Every partition π ∈ NC(n) has associated to it a permutation of {1,...,n}, which is
denotedbyP ,andisdefinedbythefollowingprescription: foreveryblockB = {b ,...,b }
π 1 m
of π, with b < ··· < b , one creates a cycle of P by putting
1 m π
P (b ) = b ,...,P (b )= b ,P (b )= b .
π 1 2 π m−1 m π m 1
Note that in the particular case when π = 0 we have that P is the identical permutation
n 0n
of {1,...,n}, while for π = 1 we have that P is the cycle 1 7→ 2 7→ ··· 7→ n7→ 1.
n 1n
3oTheKrewerascomplementation mapisaspecialorder-reversingbijectionK : NC(n)→
NC(n). In this paper we will use its description in terms of permutations associated to
non-crossing partitions: for π ∈ NC(n), the Kreweras complement of π is the partition
K(π) ∈ NC(n) uniquely determined by the fact that its associated permutation is
P = P−1P . (2.1)
K(π) π 1n
6
Formula(2.1)canbeextendedinordertocovertheconceptofrelative Kreweras complement
of π in ρ, for π,ρ ∈ NC(n) such that π ≤ ρ. This is the partition in NC(n), denoted by
K (π), uniquely determined by the fact that the permutation associated to it is
ρ
P = P−1P . (2.2)
Kρ(π) π ρ
Clearly, the Kreweras complementation map K from (2.1) is the relative complementation
with respect to the maximal element 1 of NC(n).
n
The formulas (2.1), (2.2) do not follow exactly the original approach used by Kreweras
in [8], but are easily seen to be equivalent to it (see e.g. [11], Exercise 18.25 on p. 301).
Remark 2.2. In this remark we record a few facts about relative Kreweras complements
that will be used later on in the paper.
1o For π ≤ ρ in NC(n), the number of blocks of the relative Kreweras complement
K (π) is
ρ
|K (π)| = n+|ρ|−|π| (2.3)
ρ
(see e.g. [11], Exercise 18.23 on p. 300).
2o For a fixed partition ρ∈ NC(n), the relative Kreweras complements K (π) of parti-
ρ
tions π ≤ ρ can be obtained by taking “separate Kreweras complements” inside each block
of ρ. More precisely, let us write explicitly ρ= {B ,...,B }. It is easy to see that one has
1 q
a natural poset isomorphism
{π ∈ NC(n)| π ≤ ρ} ∋ π 7→ (π ,...,π )∈ NC(|B |)×···×NC(|B |) (2.4)
1 q 1 q
where for every 1 ≤ j ≤ q the partition π ∈ NC(|B |) is obtained by restricting π
j j
to B and by re-denoting the elements of B , in increasing order, so that they become
j j
1,2,...,|B |. The above statement about taking separate complements inside each block
j
of ρ then amounts to the fact that for π ≤ ρ in NC(n) we have the implication
π 7→ (π ,...,π ) ⇒ K (π) 7→ (K(π ),...,K(π )) , (2.5)
1 q ρ 1 q
(cid:16) (cid:17) (cid:16) (cid:17)
where K(π ),...,K(π ) are Kreweras complements calculated in NC(|B |),...,NC(|B |),
1 q 1 q
respectively. For a discussion of this, see pp. 288-290 in Lecture 18 of [11].
3o From part 2o of this remark it is immediate that, for a fixed ρ ∈ NC(n), the map
π 7→ K (π) is an order-reversing bijection from {π ∈ NC(n) | π ≤ ρ} onto itself. On the
ρ
other hand, it can be shown that
π ≤ ρ ≤ ρ in NC(n) ⇒ K (π) ≤ K (π) (2.6)
1 2 ρ1 ρ2
(see Exercise 18.25.1 on p. 301 of [11]). Hence the expression “K (π)” is increasing as a
ρ
function of ρ, and is decreasing as a function of π.
The particular case when ρ = 1 in (2.6) gives us that
2 n
π ≤ ρ in NC(n) ⇒ K (π) ≤ K(π). (2.7)
ρ
Thus for every π ∈ NC(n) it makes sense to define a map
{ρ ∈ NC(n)|ρ≥ π} → {σ ∈ NC(n)| σ ≤ K(π)}
(2.8)
(cid:26) ρ 7→ Kρ(π).
7
This map turns out to be bijective, and a helpful fact for calculating its inverse is that
ρ≥ π, K (π) = σ ⇒ K (σ) = K(ρ) (2.9)
ρ K(π)
(cid:16) (cid:17)
(see Lemma 18.9 and Remark 18.10 on p. 291 of [11]). Finally, let us also record here the
fact that the whole family of bijections in (2.8) (corresponding to the various partitions
π ∈ NC(n)) can be consolidated into one bijection,
{(π,ρ) | π,ρ ∈NC(n), π ≤ ρ} → {(π,σ) |π,σ ∈ NC(n), σ ≤ K(π)}
(2.10)
(cid:26) (π,ρ) 7→ (π,Kρ(π)).
2B. Power series
Here we review a few relevant notations and facts about series in non-commuting inde-
terminates, and in particular about multivariable R-transforms.
Notation 2.3. Let k be a positive integer. As already mentioned in the introduction, we
use the notation [k]∗ for the set of all words of finite length over the alphabet {1,...,k}:
[k]∗ := ∪∞ {1,...,k}n. (2.11)
n=0
The length of a word w ∈ [k]∗ will be denoted by |w|. (In (2.11) we followed the standard
procedure of also including into [k]∗ a unique word φ with |φ|= 0.)
Definition 2.4. (Series and their coefficients.) Let k be a positive integer.
1o We will use the notation C hhz ,...,z ii for the set of power series with complex co-
0 1 k
efficients andwithvanishingconstantterminthenon-commutingindeterminatesz ,...,z .
1 k
The general form of a series f ∈ C hhz ,...,z ii is thus
0 1 k
∞ k
f(z ,...,z )= α z ···z = α z , (2.12)
1 k (i1,...,in) i1 in w w
nX=1 i1,.X..,in=1 wX∈[k]∗,
|w|≥1
where the coefficients α are from C and where, same as in the introduction, we write in
w
short z := z ···z for w = (i ,...,i ) ∈ {1,...,k}n, n ≥ 1.
w i1 in 1 n
2o For every word w ∈ [k]∗ with |w| ≥ 1 we will denote by
Cf : C hhz ,...,z ii → C
w 0 1 k
the linear functional which extracts the coefficient of z in a series f ∈ C hhz ,...,z ii.
w 0 1 k
Thus for f written as in Equation (2.12) we have Cf (f)= α .
w w
3o Suppose we are given a positive integer n, a word w = (i ,...,i ) ∈ {1,...,k}n and
1 n
a partition π ∈ NC(n). We define a (generally non-linear) functional
Cf :C hhz ,...,z ii → C,
w;π 0 1 k
as follows. For every block B = {b ,...,b } of π, with 1 ≤ b < ··· < b ≤ n, let us use
1 m 1 m
the notation
w|B = (i ,...,i )|B := (i ,...,i ) ∈ {1,...,k}m. (2.13)
1 n b1 bm
8
Then we define
Cf (f) := Cf (f), ∀f ∈ C hhz ,...,z ii. (2.14)
w;π w|B 0 1 k
B bloYck of π
(For example if w = (i ,...,i ) is a word of length 5 and if π = {{1,4,5},{2,3}} ∈
1 5
NC(5), then the above formula comes to Cf (f) = Cf (f) · Cf (f),
(i1,i2,i3,i4,i5);π (i1,i4,i5) (i2,i3)
f ∈C hhz ,...,z ii.)
0 1 k
Definition2.5. LetµbeadistributioninD (k)(thatis,µ :ChX ,...,X i → Cisalinear
alg 1 k
functional such that µ(1) = 1). The R-transform of µ is the series R ∈ C hhz ,...,z ii
µ 0 1 k
uniquely determined by the requirement that for every n ≥ 1 and every 1 ≤ i ,...,i ≤ k
1 n
one has
µ(X ···X )= Cf (R ). (2.15)
i1 in (i1,...,in);π µ
X
π∈NC(n)
Remark 2.6. It is easy to see that Equation (2.15) does indeed determine a unique series
in C hhz ,...,z ii. The coefficients of R are called the free cumulants of µ, and because
0 1 k µ
of this reason Equation (2.15) is sometimes referred to as the (free) “moment–cumulant
formula” – see Lectures 11 and 16 in [11].
Observe that, for n = 1, Equation (2.15) simply says that Cf (R )= µ(X ), 1 ≤ i ≤ k.
(i) µ i
This explains why for µ ∈ G the R-transform R was displayed in Equation (1.6) of the
k µ
introduction by having all its linear coefficients equal to 1.
An important point for the present paper is that the R-transform has a very nice be-
haviour under the operation ⊠. This is recorded in the next proposition.
Proposition 2.7. Let µ,ν be distributions in D (k), and let w be a word in [k]∗, with
alg
|w| ≥ 1. Then
Cfw Rµ⊠ν = Cfw;π Rµ ·Cfw;K(π) Rν . (2.16)
(cid:0) (cid:1) π∈XNC(n) (cid:0) (cid:1) (cid:0) (cid:1)
For the proof of Proposition 2.7 we refer to Theorem 14.4 and Proposition 17.2 of [11].
2C. Graded connected Hopf algebras
We will work with graded bialgebras over C and we will use the standard conventions
for notations regarding them (as in the monograph [15], for instance). Our review here
does not aim at generality, but just covers the specialized Hopf algebras used in the present
paper.
Notation 2.8. Let B be a graded bialgebra over C.
1o The comultiplication and counit of B will be denoted by ∆ and respectively ε (or by
∆ andε whennecessarytodistinguishB fromothergradedbialgebrasthatareconsidered
B B
at the same time). The iterations of ∆ will be denoted as ∆ℓ, ℓ ≥ 1. Thus every ∆ℓ is a
linear map from B to B⊗ℓ, where ∆1 = id (the identity map from B to B), ∆2 =∆, and for
ℓ ≥ 3 we put
∆ℓ := ∆⊗id⊗···⊗id ◦∆ℓ−1. (2.17)
(cid:0) (cid:1)
ℓ−2
| {z }
9
2o For every n ≥ 0, the vector subspace of B which consists of homogeneous elements
of degree n will be denoted by B . We thus have a direct sum decomposition B = ⊕∞ B
n n=0 n
where
B ∋ 1 (the unit of B), ε| = 0, ∀n ≥ 1,
0 B and Bn
(cid:26) Bm·Bn ⊆ Bm+n, ∀m,n ≥ 0, (cid:26) ∆(Bn)⊆ ⊕ni=0Bi⊗Bn−i, ∀n≥ 0.
If the space B of homogeneous elements of degree 0 is equal to C1 then we say that the
0 B
graded bialgebra B is connected.
Remark 2.9. (The convolution algebra L(B,M).) Let B be a graded connected bialgebra,
let M be a unital algebra over C, and let L(B,M) denote the vector space of all linear
maps from B to M. For ξ,η ∈ L(B,M) one can define their convolution product, denoted
here simply as “ξη”, by the formula
ξη := Mult◦(ξ ⊗η)◦∆, (2.18)
where Mult : M⊗M → M is the linear map given by multiplication (Mult(x⊗y) = xy
for x,y ∈ M). In other words, (2.18) says that for b ∈ B with ∆(b) = n b′ ⊗b′′ one has
i=1 i i
P
n
(ξη)(b) := ξ(b′)η(b′′) ∈M. (2.19)
i i
Xi=1
When endowed with its usual vector space structure and with the convolution product,
L(B,M) becomes itself a unital algebra over C. The unit of L(B,M) is the linear map
B ∋ b 7→ ε(b)1 , which by a slight abuse of notation is still denoted as ε (same notation as
M
for the counit of B).
Let us also record here that when one considers a convolution productof ℓ ≥ 1 elements
ξ ,...,ξ ∈ L(B,M), then Equation (2.18) becomes
1 ℓ
ξ ξ ···ξ = Mult ◦(ξ ⊗···⊗ξ )◦∆ℓ, (2.20)
1 2 ℓ ℓ 1 ℓ
where ∆ℓ : B → B⊗ℓ is the ℓth iteration of the comultiplication and Mult : M⊗ℓ → M is
ℓ
given by multiplication.
For every ξ ∈ L(B,M) it makes sense to form polynomial expressions in ξ, that is,
expressions of the form
n
t ξℓ ∈ L(B,M), for n ≥ 0 and t ,t ,...,t ∈ C,
ℓ 0 1 n
Xℓ=0
where the ξℓ (0 ≤ ℓ ≤ n) are convolution powers of ξ, and we make the convention that
ξ0 := ε (the unit of L(B,M)). An important point for the present paper is that if ξ is such
that ξ(1 )= 0 then it also makes sense to define an element
B
∞
η := t ξℓ ∈ L(B,M), (2.21)
ℓ
Xℓ=0
for an arbitrary infinite sequence (t ) in C. Indeed, if ξ(1 ) = 0 then by using (2.20) and
ℓ ℓ≥0 B
the fact that ∆ respects the grading one immediately sees that ξℓ vanishes on B whenever
n
ℓ > n. Thus η from (2.21) can be defined as the unique linear map from B to M which
satisfies
N
η| = t ξℓ | , ∀N ≥ n ≥ 0. (2.22)
Bn ℓ Bn
(cid:16)Xℓ=0 (cid:17)
10