Quantum Groups with Partial Commutation
Relations
Roland Speicher
Saarland University
Saarbrücken, Germany
supported by ERC Advanced Grant
“Non-Commutative Distributions in Free Probability”
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Some New Quantum Groups
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Quantum Groups
Section 1
Quantum Groups
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Quantum Groups
What Are Quantum Groups?
are generalizations of groups G (actually, of C(G))
are supposed to describe non-classical symmetries
are Hopf algebras, with some additional structure ...
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Quantum Groups
What Are Quantum Groups?
Deformation of classical symmetries: G
Gq
quantum groups are often deformations Gq of classical groups,
depending on some parameter q, such that for q → 1, they go to the
classical group G = G1
Gq and G1 are incomparable, none is stronger than the other
I
I
G1 is supposed to act on commuting variables
Gq is the right replacement to act on q-commuting variables
Strengthening of classical symmetries: G
G+
there are situations where a classical group G has a genuine
non-commutative analogue G+ (no interpolations)
G+ is "stronger" than G:
G ⊂ G+
I
I
G acts on commuting variables
G+ is the right replacement for acting on maximally non-commuting
variables
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Quantum Groups
Orthogonal Hopf Algebras
We are interested in quantum versions of real compact matrix groups.
Think of
orthogonal matrices
or
permutation matrices
Such quantum versions are captured by the notion of orthogonal Hopf
algebra.
Definition
An orthogonal Hopf algebra is a C ∗ -algebra A, given with a system of
n2 self-adjoint generators uij ∈ A (i, j = 1, . . . , n), subject to the following
conditions:
The inverse of u = (uij ) is the transpose matrix ut = (uji ).
∆(uij ) = Σk uik ⊗ ukj defines a morphism ∆ : A → A ⊗ A.
ε(uij ) = δij defines a morphism ε : A → C.
S(uij ) = uji defines a morphism S : A → Aop .
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Quantum Groups
Orthogonal Hopf Algebras
Definition
An orthogonal Hopf algebra is a C ∗ -algebra A, given with a system of
n2 self-adjoint generators uij ∈ A (i, j = 1, . . . , n), subject to the following
conditions:
The inverse of u = (uij ) is the transpose matrix ut = (uji ).
∆(uij ) = Σk uik ⊗ ukj defines a morphism ∆ : A → A ⊗ A.
ε(uij ) = δij defines a morphism ε : A → C.
S(uij ) = uji defines a morphism S : A → Aop .
These are compact quantum groups in the sense of Woronowicz.
In the spirit of non-commutative geometry, we are thinking of
A = C(G+ ) as the continuous functions, generated by the coordinate
functions uij , on some (non-existing) quantum group G+ , replacing a
classical group G.
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Quantum Groups
Theorem
The dual C(On ) of the classical orthogonal group is the universal
commutative unital C ∗ -algebra generated by commuting selfadjoint uij
(i, j = 1, . . . , n) subject to the relation: u = (uij )ni,j=1 is an orthogonal
matrix; i.e., for all i, j we have
n
X
uik ujk = δij
and
k=1
n
X
uki ukj = δij
k=1
Definition (Quantum Orthogonal Group On+ (Wang 1995))
The quantum orthogonal group Ao (n) = C(On+ ) is the universal unital
C ∗ -algebra generated by selfadjoint uij (i, j = 1, . . . , n) subject to the
relation: u = (uij )ni,j=1 is an orthogonal matrix; i.e., for all i, j we have
n
X
uik ujk = δij
k=1
Roland Speicher (Saarland University)
and
n
X
uki ukj = δij
k=1
Some New Quantum Groups
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Quantum Groups
Theorem
The dual C(Sn+ ) of the classical permutation group is the universal
commutative unital C ∗ -algebra generated by commuting uij
(i, j = 1, . . . , n) subject to the relations
u2ij = uij = u∗ij for all i, j = 1, . . . , n
each row and column of u = (uij )ni,j=1 is a partition of unity:
n
n
X
X
uij = 1
and
uij = 1
j=1
i=1
Definition (Quantum Permutation Group Sn+ (Wang 1998))
The quantum permutation group As (n) = C(Sn+ ) is the universal unital
C ∗ -algebra generated by uij (i, j = 1, . . . , n) subject to the relations
u2ij = uij = u∗ij for all i, j = 1, . . . , n
n
X
uij = 1
and
n
X
j=1
i=1
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uij = 1
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Quantum Groups
Actions of Quantum Groups
Definition
The action of a matrix quantum group u = (uij )ni,j=1 on a vector of
operators x = (x1 , . . . , xn ) is given by
x 7→ y = u x,
(x1 , . . . , xn ) 7→ (y1 , . . . , yn )
where
i.e.
yi =
n
X
uij ⊗ xj
j=1
Invariance under such actions corresponds to interesting geometric or
probabilistic properties
Example (classical examples)
the classical sphere is invariant under the orthogonal group
distribution of classical i.i.d. (independent, identically distributed)
random variables is invariant under permutations of the variables
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Quantum Groups
Actions of Quantum Groups
Definition
The action of a matrix quantum group u = (uij )ni,j=1 on a vector of
operators x = (x1 , . . . , xn ) is given by
x 7→ y = u x,
(x1 , . . . , xn ) 7→ (y1 , . . . , yn )
where
i.e.
yi =
n
X
uij ⊗ xj
j=1
Invariance under such actions corresponds to interesting geometric or
probabilistic properties
Example (non-commutative examples)
the “free” sphere is invariant under the quantum orthogonal group
non-commutative distribution of f.i.d. (free, identically distributed)
random variables is invariant under quantum permutations
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Quantum Groups
de Finetti Theorems
Theorem (de Finetti 1931, Hewitt+Savage 1955)
The following are equivalent for an infinite sequence of classical random
variables:
the sequence is exchangeable (i.e., invariant under each Sn )
the sequence is independent and identically distributed with respect to
the conditional expectation E onto the tail σ-algebra of the sequence
Theorem (Köstler+Speicher 2009)
The following are equivalent for an infinite sequence of non-commutative
random variables:
the sequence is quantum exchangeable (i.e., invariant under each Sn+ )
the sequence is free and identically distributed with respect to the
conditional expectation E onto the tail-algebra of the sequence
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Mixtures of Classical and Free Independence
Section 2
Mixtures of Classical and Free Independence
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Mixtures of Classical and Free Independence
History
Młotkowski (2004): Λ-freeness
Wysoczanksi + Speicher (2016): ε-independence
Speicher + Weber (2016): new corresponding quantum groups
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Mixtures of Classical and Free Independence
Non-commutative probability spaces given by groups
(CG, τ ) or (L(G), τ )
G
nc (W ∗ -) probability space
where
X
τ(
αg g) = α1
or τ (x) = hxδ1 , δ1 i.
g
freeness/independence
=
ˆ
basic group constructions:
free/direct product
?i∈I Gi , ×i∈I Gi
↑
only “universal” unital constructions
which are uniform in i ∈ I
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Mixtures of Classical and Free Independence
More General, Non-Homogeneous, Group
Constructions
direct product =
free product commutation relations between all Gi
Generalize this to : partial commutation relations between groups
Fix symmetric matrix (εij )i,j∈I = (Λij )i,j∈I with 0/1 entries, with the idea
that for i 6= j
εij = εji = 1
=
ˆ
[Gi , Gj ] = 0
εij = εji = 0
=
ˆ
no relation
[εii is unspecified or sometimes it is convenient to put εii = 0.]
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Mixtures of Classical and Free Independence
Definition (Green 1990)
Let Gi (i ∈ I) be groups. The graph product ?ε Gi (corresponding to
graph with adjacency matrix ε) is
?i∈I Gi Gi , Gj (i 6= j) commute whenever εij = 1
Example
If Gi Z, then ?ε Gi has various names
right angled Artin group
free partially commutative group
trace group
etc
also: Cartier-Foata monoid, trace monoid, . . .
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Mixtures of Classical and Free Independence
Definition (Młotkowski 2004)
Subalgebras (Ai )i∈I are ε-independent in (A, ϕ) if
[Ai , Aj ] = 0 for all i 6= j with εij = 1
Consider aj ∈ Ai(j) (i = 1, . . . , n) with
I
I
ϕ(aj ) = 0 ∀j
i(j) 6= i(j + 1) modulo commutation relations
Then ϕ(a1 · · · an ) = 0
“modulo commutation relations” means: if there are k < l with i(k) = i(l),
· · · ak · · ·
· · · al · · ·
then there exists p with k < p < l such that i(k) 6= i(p) 6= i(l) and
εi(k)i(p) = 0
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Mixtures of Classical and Free Independence
Definition (Młotkowski 2004)
Subalgebras (Ai )i∈I are ε-independent in (A, ϕ) if
[Ai , Aj ] = 0 for all i 6= j with εij = 1
Consider aj ∈ Ai(j) (i = 1, . . . , n) with
I
I
ϕ(aj ) = 0 ∀j
i(j) 6= i(j + 1) modulo commutation relations
Then ϕ(a1 · · · an ) = 0
Example of mixed moments (can be calculated via centering)
For x ∈ Ai , y ∈ Aj , i 6= j we have
ϕ(xyx) = ϕ(x2 )ϕ(y)
always
(
ϕ(x2 )ϕ(y 2 ),
εij = 1
ϕ(xyxy) =
2
2
2
2
2
2
ϕ(x )ϕ(y) + ϕ(x) ϕ(y ) − ϕ(x) ϕ(y) , εij = 0
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Mixtures of Classical and Free Independence
Definition (Młotkowski 2004)
Subalgebras (Ai )i∈I are ε-independent in (A, ϕ) if
[Ai , Aj ] = 0 for all i 6= j with εij = 1
Consider aj ∈ Ai(j) (i = 1, . . . , n) with
I
I
ϕ(aj ) = 0 ∀j
i(j) 6= i(j + 1) modulo commutation relations
Then ϕ(a1 · · · an ) = 0
Theorem (Młotkowski)
There is a ε-product construction, which has all nice properties.
Proposition (Speicher+Wysoczanski)
The subalgebras CGi are ε-independent in the graph product group algebra
C(?ε Gi ) with respect to the canonical trace τ .
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Mixtures of Classical and Free Independence
Theorem (Speicher+Wysoczanski)
Let (Ai )i∈I be ε-independent. Let aj ∈ Ai(j) . Then
ϕ(a1 · · · an ) =
X
kπ (a1 , . . . , an ),
π∈N C ε [i(1),...,i(n)]
where
kπ is the product of corresponding free cumulants, according to block
structure of π
with i = (i(1), . . . , i(n)) we have
N C ε [i] := {π ∈ P(n) | π ≤ ker i, and π is (ε, i)-noncrossing}
· · · p1 · · · q1 · · · p2 · · · q2 · · ·
then εi(p1 )i(q1 ) = 1
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Mixtures of Classical and Free Independence
Canonical Question
Question
What are quantum symmetries behind ε-independence?
More generally: What are quantum symmetries going with partial
commutation relations?
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ε-Spheres and Its Symmetries
Section 3
ε-Spheres and Its Symmetries
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ε-Spheres and Its Symmetries
Quantum Spheres (Real Versions)
Quantum spheres are given by deformations of the classical sphere
Definition (classical sphere)
C(SRn−1 )
∗
= C (x1 , . . . , xn | xi =
x∗i ,
n
X
x2i = 1, xi xj = xj xi ∀i, j}
i=1
by deformations of the commutation relations as well as the sphere
relation by some parameters (Podlés 1987, Connes+Dubois-Violette
2002, Connes+Landi 2001, etc)
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ε-Spheres and Its Symmetries
Quantum Spheres (Real Versions)
Quantum spheres are given by deformations of the classical sphere
Definition (classical sphere)
C(SRn−1 )
∗
= C (x1 , . . . , xn | xi =
x∗i ,
n
X
x2i = 1, xi xj = xj xi ∀i, j}
i=1
by dropping the commutation relation or replacing it by some other
“rigid” relation, but keeping the sphere relation (Banica+Goswami
2010, Banica 2015)
Definition (free sphere, Banica+Goswami 2010)
n−1
C(SR,+
)
∗
:= C (x1 , . . . , xn | xi =
x∗i ,
n
X
x2i = 1}
i=1
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ε-Spheres and Its Symmetries
The ε-Sphere (and Its Symmetries)
Definition (Speicher+Weber 2016)
For given ε = (εij )ni,j=1 , the ε-sphere is defined by
n−1
C(SR,ε
) := C ∗ (x1 , . . . , xn | xi = x∗i ,
n
X
x2i = 1, xi xj = xj xi if εij = 1}
i=1
What are the symmetries?
We have
n−1
n−1
C(SRn−1 ) ← C(SR,ε
) ← C(SR,+
)
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ε-Spheres and Its Symmetries
The ε-Sphere (and Its Symmetries)
Definition (Speicher+Weber 2016)
For given ε = (εij )ni,j=1 , the ε-sphere is defined by
n−1
C(SR,ε
) := C ∗ (x1 , . . . , xn | xi = x∗i ,
n
X
x2i = 1, xi xj = xj xi if εij = 1}
i=1
What are the symmetries?
We have
n−1
n−1
SRn−1 ⊂ SR,ε
⊂ SR,+
and
SRn−1 is invariant under the orthogonal group On
n−1
SR,+
is invariant under the quantum orthogonal group On+
n−1
What is the invariance quantum group for the ε-sphere SR,ε
?
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ε-Spheres and Its Symmetries
Symmetries of ε-Sphere: ε-Orthogonal Group
Theorem (Speicher+Weber)
n−1
The maximal quantum group acting on the ε-sphere SR,ε
is given by the
ε
ε-orthogonal quantum group On , which is defined by
C(Onε ) = C ∗ (uij (i, j = 1, . . . , n) | u∗ij = uij , u = (uij ) is orthogonal, Rε }
Definition (relations Rε )


ujl uik , eij = 1, ekl = 1
uik ujl = ujk uil , eij = 1, ekl = 0


uil ujk , eij = 0, ekl = 1
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ε-Spheres and Its Symmetries
n−1
Onε acts on SR,ε
means that
n−1
if (x1 , . . . , xn ) ∈ C(SR,ε
)
and y = u x, i.e.,
yi =
X
uik ⊗ xk
k
n−1
then (y1 , . . . , yn ) ∈ C(SR,ε
)
Note that in general
On 6⊂ Onε ⊂ On+
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ε-Independence and Its Symmetries
Section 4
ε-Independence and Its Symmetries
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ε-Independence and Its Symmetries
The ε-Permutation Group
Definition (Speicher+Weber)
We set
+
Snε := SnRε ,
i.e.
C(Snε ) = C ∗ (uij (i, j = 1, . . . , n) | u∗ij = uij = u2ij , Rε ,
X
X
uik = 1 =
ukj ∀i, j}
k
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k
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ε-Independence and Its Symmetries
Definition (Speicher+Weber)
C(Snε ) = C ∗ (uij (i, j = 1, . . . , n) | u∗ij = uij = u2ij , Rε ,
X
X
uik = 1 =
ukj ∀i, j}
k
k
Theorem (Speicher+Weber)
1) This is the same as replacing Rε by


ujl uik , eij = 1, ekl = 1
uik ujl = 0,
eij = 1, ekl = 0


0,
eij = 0, ekl = 1
2) Hence, C(Snε ) is the quantum automorphism group of Bichon (2003) of
the graph with adjacency matrix ε.
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ε-Independence and Its Symmetries
Invariance of ε-Independence Under
ε-Permutations
Theorem (Speicher+Weber)
Let x1 , . . . , xn be ε-independent and identically distributed. Then their
distribution is invariant under Snε , i.e., with (uij )ni,j=1 ∈ Snε :
ϕ(xj(1) · · · xj(k) ) =
n
X
ϕ(xi(1) · · · xi(k) )ui(1)j(1) · · · ui(k)j(k)
i(1),...,i(k)=1
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ε-Independence and Its Symmetries
Proof.
Recall
X
ϕ(xi(1) · · · xi(k) ) =
X
kπ (xi(1) , . . . , xi(k) ) =
π∈N C ε [i]
δ(π ∈ N C ε [i]) · kπ
π∈P(k)
Hence
n
X
ϕ(xi(1) · · · xi(k) )ui(1)j(1) · · · ui(k)j(k)
i(1),...,i(k)=1

=
X
kπ 
π∈P(k)

X
δ(π ∈ N C ε [i]) · ui(1)j(1) · · · ui(k)j(k) 
i(1),...,i(k)
{z
|
=δ(π∈N C ε [j])
}
= ϕ(xj(1) · · · xj(k) )
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ε-Independence and Its Symmetries
Summary
n−1
Onε = quantum symmetry of the ε-sphere SR,ε
Snε = quantum symmetry of ε-independence
n−1
C(SR,ε
) := C ∗ (x1 , . . . , xn | xi = x∗i ,
P
i
x2i = 1, xi xj = xj xi , if εij = 1)
C(Onε ) := C ∗ (uij , i, j = 1, . . . , n | uij = u∗ij , u is orthogonal, Rε )
P
P
C(Snε ) := C ∗ (uij | uij = u∗ij = u2ij , k uik = k ukj = 1 ∀i, j and R̊ε )
 ε
R



u u
jl ik
uik ujl =

u
jk uil



uil ujk
R̊ε
ujl uik
0
0
if εij = 1 and εkl = 1
if εij = 1 and εkl = 0
if εij = 0 and εkl = 1
Speicher,Weber: Quantum groups with partial commutation relations, 1603.09192
Speicher,Wysoczanski: Mixtures of classical and free independence, 1603.08758
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