Free probability and planar algebras
Stephen Curran
MIT
AMS Joint Math Meetings
Special Session on Progress in Free Probability and Free Analysis
Stephen Curran (MIT)
Free probability and planar algebras
January 11, 2013
1 / 14
The polynomial planar algebra
Let Pm = spanhXi1 · · · Xim i ⊂ ChX1 , . . . , Xn i.
Stephen Curran (MIT)
Free probability and planar algebras
January 11, 2013
2 / 14
The polynomial planar algebra
Let Pm = spanhXi1 · · · Xim i ⊂ ChX1 , . . . , Xn i.
Planar tangles act on P = (Pm )m≥0 by “contracting indices”:
T =
Stephen Curran (MIT)
Free probability and planar algebras
January 11, 2013
2 / 14
The polynomial planar algebra
Let Pm = spanhXi1 · · · Xim i ⊂ ChX1 , . . . , Xn i.
Planar tangles act on P = (Pm )m≥0 by “contracting indices”:
T =
ZT : P6 ⊗ P4 → P6
ZT (Xi1 · · · Xi6 ⊗ Xj1 · · · Xj4 ) =
Stephen Curran (MIT)
Free probability and planar algebras
January 11, 2013
2 / 14
The polynomial planar algebra
Let Pm = spanhXi1 · · · Xim i ⊂ ChX1 , . . . , Xn i.
Planar tangles act on P = (Pm )m≥0 by “contracting indices”:
i5
i4
T =
i6
i3
i1
i2
j3
j4
j2
j1
ZT : P6 ⊗ P4 → P6
ZT (Xi1 · · · Xi6 ⊗ Xj1 · · · Xj4 ) =
Stephen Curran (MIT)
Free probability and planar algebras
January 11, 2013
2 / 14
The polynomial planar algebra
Let Pm = spanhXi1 · · · Xim i ⊂ ChX1 , . . . , Xn i.
Planar tangles act on P = (Pm )m≥0 by “contracting indices”:
i6
j4
i5
i5
i4
T =
i6
i3
i1
i2
j3
j4
j2
j1
i4
j1
j2
ZT : P6 ⊗ P4 → P6
ZT (Xi1 · · · Xi6 ⊗ Xj1 · · · Xj4 ) = δi1 i2 δi3 j3 · Xi5 Xi6 Xj4 Xj1 Xj2 Xi4
Stephen Curran (MIT)
Free probability and planar algebras
January 11, 2013
2 / 14
Temperley-Lieb planar algebra
Temperley-Lieb planar algebra: spanned by non-crossing pairings
TL3 = span
,
,
,
,
Embedding TL ⊂ ChX1 , . . . , Xn i:
7→
P
Xi2 ,
7→
P
Xi Xj Xj Xk Xk Xi
TL is a planar subalgebra of ChX1 , . . . , Xn i.
Loop parameter:
δ=
=
X
i,j
Stephen Curran (MIT)
Xj Xj
=
X
Xi Xi
Free probability and planar algebras
δij = n
i,j
January 11, 2013
3 / 14
Diagrammatic algebra structures
P = ChX1 , . . . , Xn i.
Multiplication ∧0 on A0 = ChX1 , . . . , Xn i:
···
Xi1 · · · Xim
···
Xim+1 · · · Xim+k
= Xi1 · · · Xim+k
P2 with multiplication ∧1 :
Xi
Xj
Xk
Xl
= δjk ·
Xi
Xl
Isomorphic to Mn (C).
Extend ∧1 to A1 = {Xi aXj : a ∈ A}:
(Xi aXj ) ∧1 (Xk bXl ) =
i a j
k b l
(A1 , ∧1 ) is isomorphic to A0 ⊗ Mn (C).
Stephen Curran (MIT)
Free probability and planar algebras
January 11, 2013
4 / 14
Voiculescu trace
A0 = ChX1 , . . . , Xn i.
X1 , . . . , Xn free semicircular family with respect to τ0 :
P
τ0 (a) =
TL
Loopless diagrams
a
τ1 = τ0 ⊗ tr on A1 ' A0 ⊗ Mn (C):
P
τ1 (a) =
δ −1
TL
a
Trace-preserving inclusion A0 ⊂ A1 :
a
Stephen Curran (MIT)
7→
a
Free probability and planar algebras
January 11, 2013
5 / 14
Planar algebra subfactors
Let P = (Pm )m≥0 be a planar algebra.
L
A0 (P) = m≥0 with multiplication ∧0 .
L
A1 (P) = m≥2 Pm with multiplication ∧1 .
Voiculescu trace: τk : Ak (P) → C, k = 0, 1.
Theorem (Guionnet-Jones-Shlyakhtenko ’08)
τ0 , τ1 are positive, faithful tracial states. The GNS completions
M0 (P) ⊂ M1 (P) are II1 factors. When P is finite-depth, M0 ' LFt with
t = 1 + I (δ − 1) where I is the global index of P.
Recovers famous result of Popa ’94 that every planar algebra is the
standard invariant of a subfactor.
Stephen Curran (MIT)
Free probability and planar algebras
January 11, 2013
6 / 14
Symmetric enveloping algebra of planar algebra subfactors
A0 Aop
0 =
L
s,t
Ps+t with multiplication:
x ∧y =
y
x
op
A0 ⊗ Aop
0 ⊂ A0 A0 :
x ⊗ y op 7→
x
y
op
P = ChX1 , . . . Xn i ⇒ A0 ⊗ Aop
0 = A A0 . For P = TL:
∈
/ A0 (P) ⊗ Aop
0 (P)
Stephen Curran (MIT)
Free probability and planar algebras
January 11, 2013
7 / 14
Symmetric enveloping algebra of planar algebra subfactors
Define τ0 τ0 on A0 Aop
0 by
P
τ0 τ0 (x) =
TL
x
P
TL
Theorem (C.-Jones-Shlyakhtenko ’11)
τ0 τ0 is a positive, faithful trace. The GNS completion
M0 ⊗ M0op ⊂ M0 M0op is isomorphic to Popa’s symmetric enveloping
inclusion.
P finite-depth: Popa’s symmetric enveloping inclusion is isomorphic
to Ocneanu’s asymptotic inclusion, a subfactor analogue of Drinfeld’s
quantum double construction.
C. ’12: Description of planar algebra of asymptotic inclusion when P
is finite-depth.
Stephen Curran (MIT)
Free probability and planar algebras
January 11, 2013
8 / 14
Free analysis on planar algebras
A0 = ChX1 , . . . , Xn i
An0 = {~a = (a1 , . . . , an ) : a ∈ A0 }:
~a =
n
X
Xi
ai
⇔ ai = Xi
~a
i=1
Cyclic Gradient: D : A0 → An0
D(a) =
X
a
k
k
Well-defined on A0 (P) for any planar algebra.
Stephen Curran (MIT)
Free probability and planar algebras
January 11, 2013
9 / 14
Free analysis on planar algebras
A0 = ChX1 , . . . , Xn i
Free difference quotient: ∂i : A0 → A0 ⊗ Aop
0
∂i (a) =
X
a
Xi
k
k
Free Jacobian: J : An0 → Mn (A0 ⊗ Aop
0 )
J (a) =
X
a
k
k
Well-defined on A0 (P) for any planar algebra.
Stephen Curran (MIT)
Free probability and planar algebras
January 11, 2013
10 / 14
Free Gibbs states on planar algebras
For V = V ∗ ∈ A0 (P), the free Gibbs state τV is defined by
J ∗ (1) = DV (adjoint computed relative to L2 (τV )).
Guionnet-MaurelP
Segala ’06: For P = ChX1 , . . . , Xn i, V sufficiently
1
1
close to 2 ∪ = 2 Xi2 , τV exists and is given by
Z
1
τV (P) = lim
tr(P(X ))e −N Tr[V (X )] dX1 · · · dXn .
N→∞ ZN (V ) (M sa )n
N
Theorem (Guionnet-Jones-Shlyakhtenko-Zinn Justin ’10)
For any planar algebra P and V = V ∗ ∈ A0 (P) sufficiently close to 12 ∪,
τV exists as a limit of random matrix models.
Stephen Curran (MIT)
Free probability and planar algebras
January 11, 2013
11 / 14
Free monotone transport
Theorem (Guionnet-Shlyakhtenko ’12)
Let X1 , . . .P
, Xn be a free semicircular system in (M, τ ). For V sufficiently
close to 12 Xi2 there are Y1 , . . . , Yn satisfying:
Yi = Fi (X1 , . . . , Xn ) for some n.c. power series Fi and
C ∗ (X1 , . . . , Xn ) = C ∗ (Y1 , . . . , Yn ).
Transport: τ (P(Y )) = τV (P) for any n.c. polynomial P.
Monotone: Yi = Di (G ) for some G ∈ C ∗ (X1 , . . . , Xn ) and J Y is
positive-definite.
In particular, W ∗ (τV ) ' LFn (solving a conjecture of Voiculescu).
Stephen Curran (MIT)
Free probability and planar algebras
January 11, 2013
12 / 14
Free monotone transport and planar algebras
For Y ∈ A1 (P) define evY : A0 → A0 ,
Y
···
Y
evY (P) =
Y
P
For P = ChX1 , . . . , Xn i, evY (P) = P(Y ), Y = (Y1 , . . . , Yn ).
Theorem (C.-Dabrowski-Shlyakhtenko ’13)
Let τ be the Voiculescu trace on a finite-depth planar algebra P. For
V = V ∗ ∈ A0 (P) sufficiently close to (1/2) · ∪, there is a Y satisfying:
Y ∈ C ∗ (A1 ) and W ∗ ({evY (P) : P ∈ A0 }) = W ∗ (A0 ).
Transport: τ (evY (P)) = τV (P) for P ∈ A0 .
Monotone: Y = DG for some G ∈ W ∗ (A0 ) and J Y is positive-definite.
In particular, W ∗ (τV ) ' LFt , t = 1 + I (δ 2 − 1).
Stephen Curran (MIT)
Free probability and planar algebras
January 11, 2013
13 / 14
Thank you!
Stephen Curran (MIT)
Free probability and planar algebras
January 11, 2013
14 / 14