Hopf algebra approach to and S-transform
Mitja Mastnak
1 Saint
Alexandru Nica
2
Mary’s University, Halifax, Canada
2 University
Alexandru Nica
1
of Waterloo, Canada
Hopf algebra approach to and S-transform
Main goal of the talk is to advertise connection between:
(1) multiplication of free random variables
(operation , the S-transform – in algebraic framework)
and
(2) symmetric functions.
Alexandru Nica
Hopf algebra approach to and S-transform
Multiplication of free random variables (I)
Will use an algebraic framework:
• Denote Dalg := {µ : C[X ] → C | µ linear, µ(1) = 1}.
• Have operation on Dalg which follows the multiplication of
free elements in a non-commutative probability space (A, ϕ).
(If a is free from b in (A, ϕ), with a having distribution µ and b
having distribution ν, then the product ab has distribution µ ν.)
• Can write concrete formulas for moments of µ ν, as
polynomials of the moments of µ, ν. In particular have
(µ ν)(X ) = µ(X ) · ν(X ).
• Put G := {µ ∈ Dalg | µ(X ) = 1}. Then (G, ) is a commutative
group.
Natural Question: Identify what is the group (G, ).
Alexandru Nica
Hopf algebra approach to and S-transform
Multiplication of free random variables (II)
Natural Question: Identify what is the group (G, ).
This question solved by Voiculescu (1987) by using S-transform.
Let F be the set of (formal) power series of the form
f (z) = 1 +
∞
X
γn z n ,
with γn ’s in C.
n=1
Every µ ∈ G has an S-stransform Sµ ∈ F, and
Sµν = Sµ · Sν , ∀ µ, ν ∈ G.
One gets a group isomorphism (G, ) ' (F, ·), where operation
on F is plain multiplication of series.
<−1> (z), where M is
[Concrete formula for Sµ is Sµ (z) = 1+z
µ
z Mµ
moment series and inverse is with respect to operation of
composing power series.]
Alexandru Nica
Hopf algebra approach to and S-transform
Multiplication of free random variables (III)
Remark. can also be addressed by using another important
transform of free probability, the R-transform. For µ ∈ G, the
R-transform Rµ is a series of the form
Rµ (z) =
∞
X
αn z n ,
n=1
with α1 = µ(X ) = 1. One has nice combinatorial formulas (using
summations over non-crossing partitions) for the coefficients of
Rµν in terms of the coefficients of Rµ and of Rν . This is less
efficient than the S-transform, but has the advantage that
formulas for the R-transform have straightforward extensions to
joint distributions of k-tuples of elements in a non-commutative
probability space.
Our goal is to relate (G, ) to symmetric functions, so we next
look at the algebra Sym.
Alexandru Nica
Hopf algebra approach to and S-transform
Symmetric functions I (algebra structure)
Denote Sym := algebra of symmetric polynomials in countable
family of (commuting) indeterminates (ti )∞
i=1 . Fundamental
theorem of symmetric functions says
Sym ' C[e1 , e2 , . . . , en , . . . ]
where the en ’s are the elementary symmetric functions:
e1 =
∞
X
i=1
ti , e 2 =
∞
X
1≤i<j
ti tj , . . . , en :=
X
ti1 ti2 · · · tin , . . .
1≤i1 <···<in
Denote X(Sym) := {χ : Sym → C | χ is a character} (i.e. χ is
linear and multiplicative, with χ(1) = 1). Note that χ is defined by
prescribing at will the sequence of numbers χ(en ), n ≥ 1.
Alexandru Nica
Hopf algebra approach to and S-transform
Symmetric functions II (bialgebra structure)
Algebra Sym carries a bialgebra structure, i.e. has counit
ε : Sym → C and comultiplication ∆ : Sym → Sym ⊗ Sym. These
are unital algebra homomorphisms, uniquely determined by
requirement that ε(en ) = 0 for all n ≥ 1 and that
∆(en ) =
n
X
ei ⊗ en−i , ∀ n ≥ 0 (with convention e0 := 1).
i=0
[With these added operations, Sym turns out to be a graded
connected Hopf algebra.]
Consequence of bialgebra structure: get operation of convolution
f ? g for two linear functionals f , g : Sym → C. In particular,
X(Sym) becomes a group under convolution.
Alexandru Nica
Hopf algebra approach to and S-transform
The generators yn for Sym
For the connection between and Sym, we need another set of
generators for Sym, which we call (yn )∞
n=1 . Every yn is a
homogeneous symmetric function of degree n − 1.
Low order formulas for the yn :
y1 = 1, y2 = e1 , y3 = e2 + e12 , y4 = e3 + 3e1 e2 + e13 , . . .
Comultiplication ∆(yn ) has special writing in terms of ym ’s with
m ≤ n. Low order formulas:
∆(y1 ) = 1⊗1 = y1 ⊗y1 , ∆(y2 ) = 1⊗e1 +e1 ⊗1 = y12 ⊗y2 +y2 ⊗y12 ,
∆(y3 ) = ∆(e2 ) + (∆(e1 ))2 = · · · = y13 ⊗ y3 + 3y1 y2 ⊗ y1 y2 + y3 ⊗ y13 .
General formula for yn and ∆(yn ) are in terms of non-crossing
partitions (which is what will connect the yn ’s to ). Do first a
quick review of NC (n) terminology.
Alexandru Nica
Hopf algebra approach to and S-transform
Some NC(n) terminology
• For every n ∈ N, we denote by NC (n) the set of all non-crossing
partitions of {1, . . . , n}. General partition in NC (n) is of the form
π = {V1 , . . . , Vp } (with V1 ∪ · · · ∪ Vp = {1, . . . , n} and where for
i 6= j we have that Vi ∩ Vj = ∅ and that Vi , Vj do not cross).
• NC (n) has a natural partial order by reverse refinement, which
makes it become a lattice.
• There exists an important lattice anti-isomorphism
Kr : NC (n) → NC (n), called the Kreweras complementation map.
Notation. For every n ≥ 1 and every π = {V1 , . . . , Vp } ∈ NC (n)
we denote
eπ := e|V1 | · · · e|Vp | ∈ Sym.
Alexandru Nica
Hopf algebra approach to and S-transform
The generators yn for Sym (general formula)
P
Definition. For every n ≥ 2 put yn := π∈ NC (n−1) eπ .
(E.g. y3 = e12 + e2 , sum of two terms, corresponding to the two
partitions in NC (2).) We also define y1 := 1, and we extend from
yn ’s to yπ ’s – that is, for π = {V1 , . . . , Vp } in NC (n) we denote
yπ := y|V1 | · · · y|Vp | ∈ Sym.
Remark. The formulas connecting yn ’s to en ’s can be reversed. As
a consequence, one can define characters χ ∈ X(Sym) by
prescribing at will the values χ(yn ) ∈ C, n ≥ 2 (to which we add
the condition χ(y1 ) = χ(1) = 1).
Proposition. For every n ≥ 1 we have
X
∆(yn ) =
yπ ⊗ yKr (π)
π∈NC (n)
(where Kr (π) ∈ NC (n) is the Kreweras complement of π).
Alexandru Nica
Hopf algebra approach to and S-transform
Relation betwen (G, ) and Sym
Definition.
P Let µ ∈n G be given, and consider the R-transform
Rµ (z) = ∞
n=1 αn z (where α1 = µ(X ) = 1). We denote by χµ
the character in X(Sym) uniquely determined by the requirement
that χ(yn ) = αn , ∀ n ≥ 1.
Theorem. The map G 3 µ 7→ χµ ∈ X(Sym) gives a group
isomorphism between (G, ) and ( X(Sym), ?).
Idea of Proof. One has a summation formula (from Nica-Speicher
1996) which describes explicitly the coefficients of the R-transform
Rµν in terms of the coefficients of Rµ and Rν . This matches the
formula found for ∆(yn ) on the preceding slide, and leads to
χµν = χµ ? χν .
Alexandru Nica
Hopf algebra approach to and S-transform
Generalize to joint distributions of k-tuples
Fix a positive integer k. Consider algebra ChX1 , . . . , Xk i of
polynomials in non-commuting X1 , . . . , Xk . Denote
Dalg (k) := {µ : ChX1 , . . . , Xk i → C | µ linear, µ(1) = 1}.
Have operation on Dalg (k), which follows the multiplication of
free k-tuples of elements in a non-commutative probability space.
If we put
Gk = {µ ∈ Dalg (k) | µ(Xi ) = 1, ∀ 1 ≤ i ≤ k},
then (Gk , ) is a group (non-commutative, if k ≥ 2).
Natural Question: Identify what is the group (Gk , ).
Alexandru Nica
Hopf algebra approach to and S-transform
Hopf algebra approach to (Gk , )
Natural Question: Identify what is the group (Gk , ).
The theorem shown earlier (corresponding to k = 1) obtains
(G1 , ) ' (X(Sym), ?), by using the R-transform. This can then
be extended to arbitrary (Gk , ), by using a suitable Hopf algebra
Yk instead of Sym. Use a commutative algebra of polynomials:
`
Yk = C[ yw | w ∈ Wk ], where Wk = ∪∞
`=2 {1, . . . , k}
(“words” over the alphabet {1, . . . , k}). What relates Yk to is
its comultiplication: we introduce products yw ;π for π ∈ NC (n)
and w ∈ Wk having |w | = n, then define ∆ by putting
X
∆(yw ) =
yw ;π ⊗ yw ;Kr (π) , for w ∈ Wk with |w | = n.
π∈NC (n)
Alexandru Nica
Hopf algebra approach to and S-transform
Relation betwen (Gk , ) and Yk
Definition. Let µ ∈ Gk be given, and consider the R-transform
Rµ (z1 , . . . , zk ) =
k
X
i=1
zi +
X
αw zw
w ∈Wk
(with notation zw := zi1 · · · zin for w = (i1 , . . . , in )). We denote by
χµ the character in X(Yk ) uniquely determined by the requirement
that χ(Yw ) = αw , ∀ w ∈ Wk .
Theorem. The map Gk 3 µ 7→ χµ ∈ X(Yk ) gives a group
isomorphism between (Gk , ) and ( X(Yk ), ?).
Idea of Proof. One has a summation formula which describes
explicitly the coefficients of the R-transform Rµν in terms of the
coefficients of Rµ and Rν . This matches the formula used to define
∆(yw ) and gives χµν = χµ ? χν .
Alexandru Nica
Hopf algebra approach to and S-transform
Partial linearization of on Gk , by LS-transform (I)
Remark. Every ξ ∈ X(Yk ) has a logarithm
This is the
P∞log(ξ).
1
functional η : Yk → C defined as η = − i=1 i (ε − ξ)?i (infinite
sum makes sense due to the natural grading of Yk ). Moreover, if
ξ1 , ξ2 ∈ X(Yk ) are such that ξ1 ? ξ2 = ξ2 ? ξ1 , then
log(ξ1 ? ξ2 ) = log(ξ1 ) + log(ξ2 ).
Definition. Let µ ∈ Gk be given, and let χµ ∈ X(Yk ) be defined
as on the preceding slide. The power series
X LSµ (z1 , . . . , zk ) :=
(log χµ (yw )) zw
w ∈Wk
is called the LS-transform of µ.
Remark. One can also define the coefficients of LSµ directly in
terms of the coefficients of Rµ , by using summations over chains in
the lattices NC (n).
Alexandru Nica
Hopf algebra approach to and S-transform
Partial linearization of on Gk , by LS-transform (II)
Theorem. LSµν = LSµ + LSν , for any µ, ν ∈ Gk such that
µ ν = ν µ.
Proof – immediate (from µ ν = ν µ we get that
χµ ? χν = χµν = χν ? χµ , hence that
log χµν = log χµ + log χν , and the result follows from definition
of the LS-transform).
Theorem. Suppose that k = 1, hence Gk = G1 = G. For µ ∈ G1
can consider both the S-transform Sµ (z) and the LS-transform
LSµ (z). Then LSµ (z) = −z log Sµ (z).
Proof – not immediate (need to use the connection between Rµ
and 1/Sµ , and compare that to the connection in Sym between
en ’s and the power sum symmetric functions).
Alexandru Nica
Hopf algebra approach to and S-transform
A problem: -convolution powers for k-tuples?
Definition. Let µ ∈ Gk and p > 0 be given. The convolution power
µp is the distribution ν ∈ Gk uniquely determined by the
requirement that LSν = p · LSµ .
(∗)
Notation. Denote by Gk the set of distributions µ ∈ Gk which
arise from k-tuples of selfadjoints in C ∗ -framework, in the
following sense: one can find a C ∗ -probability space (A, ϕ) and
selfadjoint elements a1 , . . . , ak ∈ A such that
µ(Xi1 · · · Xin ) = ϕ(ai1 · · · ain ), ∀ n ≥ 1 and 1 ≤ i1 , . . . , in ≤ k.
A loosely stated problem. Give interesting examples of µ and p
(∗)
with µ ∈ Gk and p > 0, such that the convolution power µp still
(∗)
belongs to Gk .
(‘Interesting’ in multi-variable sense, i.e. k ≥ 2 and µ not coming
in some obvious way from 1-dimensional distributions.)
Alexandru Nica
Hopf algebra approach to and S-transform