[PPT] - On the symmetric enveloping algebra of planar algebra subfactors PowerPoint Presentation

SLIDE 1

On the symmetric enveloping algebra of planar algebra subfactors (Joint work with V. Jones and D. Shlyakhtenko)

Stephen Curran

UCLA

Workshop on II1 factors Institut Henri Poincar´ e May 26, 2011

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 1 / 16

SLIDE 2

Planar algebras

A (subfactor) planar algebra is a sequence of finite dimensional vector spaces (Pn,±)n≥0 with an action of planar tangles.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

SLIDE 3

Planar algebras

A (subfactor) planar algebra is a sequence of finite dimensional vector spaces (Pn,±)n≥0 with an action of planar tangles.

T = D1 D2

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

SLIDE 4

Planar algebras

A (subfactor) planar algebra is a sequence of finite dimensional vector spaces (Pn,±)n≥0 with an action of planar tangles.

T = D1 D2

ZT : P3,− ⊗ P2,+ → P3,+

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

SLIDE 5

Planar algebras

A (subfactor) planar algebra is a sequence of finite dimensional vector spaces (Pn,±)n≥0 with an action of planar tangles.

T = D1 D2 S =

ZT : P3,− ⊗ P2,+ → P3,+ ZS : P0 → P2,+

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

SLIDE 6

Planar algebras

A (subfactor) planar algebra is a sequence of finite dimensional vector spaces (Pn,±)n≥0 with an action of planar tangles.

T ◦2 S = D1 S =

ZT : P3,− ⊗ P2,+ → P3,+ ZS : P0 → P2,+

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

SLIDE 7

Planar algebras

A (subfactor) planar algebra is a sequence of finite dimensional vector spaces (Pn,±)n≥0 with an action of planar tangles.

T ◦2 S = D1 S =

ZT : P3,− ⊗ P2,+ → P3,+ ZS : P0 → P2,+ ZT◦2S = ZT ◦2 ZS

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

SLIDE 8

Planar algebras

A (subfactor) planar algebra is a sequence of finite dimensional vector spaces (Pn,±)n≥0 with an action of planar tangles.

T = D1 D2 S =

ZT : P3,− ⊗ P2,+ → P3,+ ZS : P0 → P2,+ ZT◦2S = ZT ◦2 ZS Further conditions: P0 = C, ∗-structure, positivity, sphericality.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

SLIDE 9

Planar algebras

A (subfactor) planar algebra is a sequence of finite dimensional vector spaces (Pn,±)n≥0 with an action of planar tangles.

T = D1 D2 S =

ZT : P3,− ⊗ P2,+ → P3,+ ZS : P0 → P2,+ ZT◦2S = ZT ◦2 ZS Further conditions: P0 = C, ∗-structure, positivity, sphericality. Follows that there is δ ∈ {2 cos(π/n) : n ≥ 3} ∪ [2, ∞] s.t. ZT ′ = δ · ZT

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

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Planar algebras

A (subfactor) planar algebra is a sequence of finite dimensional vector spaces (Pn,±)n≥0 with an action of planar tangles.

T ′ = D1 D2 S =

ZT : P3,− ⊗ P2,+ → P3,+ ZS : P0 → P2,+ ZT◦2S = ZT ◦2 ZS Further conditions: P0 = C, ∗-structure, positivity, sphericality. Follows that there is δ ∈ {2 cos(π/n) : n ≥ 3} ∪ [2, ∞] s.t. ZT ′ = δ · ZT

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

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The polynomial planar algebra

Pk,± ⊂ CX1, X ∗

1 , . . . , Xn, X ∗ n

Pk,+ = span{Xi1X ∗

j1 · · · XikX ∗ jk}

Pk,− = span{X ∗

i1Xj1 · · · X ∗ ikXjk}

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 3 / 16

SLIDE 12

The polynomial planar algebra

Pk,± ⊂ CX1, X ∗

1 , . . . , Xn, X ∗ n

Pk,+ = span{Xi1X ∗

j1 · · · XikX ∗ jk}

Pk,− = span{X ∗

i1Xj1 · · · X ∗ ikXjk}

T =

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 3 / 16

SLIDE 13

The polynomial planar algebra

Pk,± ⊂ CX1, X ∗

1 , . . . , Xn, X ∗ n

Pk,+ = span{Xi1X ∗

j1 · · · XikX ∗ jk}

Pk,− = span{X ∗

i1Xj1 · · · X ∗ ikXjk}

T =

ZT(X ∗

i1Xj1X ∗ i2Xj2X ∗ i3Xj3 ⊗ Xk1X ∗ l1Xk2X ∗ l2) =

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 3 / 16

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The polynomial planar algebra

Pk,± ⊂ CX1, X ∗

1 , . . . , Xn, X ∗ n

Pk,+ = span{Xi1X ∗

j1 · · · XikX ∗ jk}

Pk,− = span{X ∗

i1Xj1 · · · X ∗ ikXjk}

T =

j2 i3 j3 i1 j1 i2 l2 k1 l1 k2

ZT(X ∗

i1Xj1X ∗ i2Xj2X ∗ i3Xj3 ⊗ Xk1X ∗ l1Xk2X ∗ l2) =

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 3 / 16

SLIDE 15

The polynomial planar algebra

Pk,± ⊂ CX1, X ∗

1 , . . . , Xn, X ∗ n

Pk,+ = span{Xi1X ∗

j1 · · · XikX ∗ jk}

Pk,− = span{X ∗

i1Xj1 · · · X ∗ ikXjk}

T =

j2 i3 j3 i1 j1 i2 l2 k1 l1 k2 j2 i3 k1 l1 k2 i2

ZT(X ∗

i1Xj1X ∗ i2Xj2X ∗ i3Xj3 ⊗ Xk1X ∗ l1Xk2X ∗ l2) = δj3i1δj1l2Xj2X ∗ i3Xk1X ∗ l1Xk2X ∗ i2

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 3 / 16

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The polynomial planar algebra

Pk,± ⊂ CX1, X ∗

1 , . . . , Xn, X ∗ n

Pk,+ = span{Xi1X ∗

j1 · · · XikX ∗ jk}

Pk,− = span{X ∗

i1Xj1 · · · X ∗ ikXjk}

T =

j2 i3 j3 i1 j1 i2 l2 k1 l1 k2 j2 i3 k1 l1 k2 i2

ZT(X ∗

i1Xj1X ∗ i2Xj2X ∗ i3Xj3 ⊗ Xk1X ∗ l1Xk2X ∗ l2) = δj3i1δj1l2Xj2X ∗ i3Xk1X ∗ l1Xk2X ∗ i2

Planar algebra of modulus δ = n.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 3 / 16

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Planar algebras and subfactors

Theorem (Jones ’99)

The standard invariant of any finite-index inclusion of II1 factors N ⊂ M has a planar algebra structure (with δ2 = [M : N]).

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 4 / 16

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Planar algebras and subfactors

Theorem (Jones ’99)

The standard invariant of any finite-index inclusion of II1 factors N ⊂ M has a planar algebra structure (with δ2 = [M : N]).

Theorem (Popa ’95)

Any planar algebra (λ-lattice) is the standard invariant of a finite-index inclusion of II1 factors.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 4 / 16

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Planar algebra subfactors

Let P = (Pn,±)n≥0 be a subfactor planar algebra.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 5 / 16

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Planar algebra subfactors

Let P = (Pn,±)n≥0 be a subfactor planar algebra. Guionnet-Jones-Shlyakhtenko ’08,’09: Tower of graded algebras Gr0(P) ⊂ Gr1(P) ⊂ · · · .

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 5 / 16

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Planar algebra subfactors

Let P = (Pn,±)n≥0 be a subfactor planar algebra. Guionnet-Jones-Shlyakhtenko ’08,’09: Tower of graded algebras Gr0(P) ⊂ Gr1(P) ⊂ · · · . Voiculescu trace τk : Grk(P) → C.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 5 / 16

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Planar algebra subfactors

Let P = (Pn,±)n≥0 be a subfactor planar algebra. Guionnet-Jones-Shlyakhtenko ’08,’09: Tower of graded algebras Gr0(P) ⊂ Gr1(P) ⊂ · · · . Voiculescu trace τk : Grk(P) → C. GNS completions give tower of II1 factors M0 ⊂ M1 ⊂ · · · , whose planar algebra is P. (Diagrammatic proof of Popa’s reconstruction theorem).

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 5 / 16

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Planar algebra subfactors

Let P = (Pn,±)n≥0 be a subfactor planar algebra. Guionnet-Jones-Shlyakhtenko ’08,’09: Tower of graded algebras Gr0(P) ⊂ Gr1(P) ⊂ · · · . Voiculescu trace τk : Grk(P) → C. GNS completions give tower of II1 factors M0 ⊂ M1 ⊂ · · · , whose planar algebra is P. (Diagrammatic proof of Popa’s reconstruction theorem). If P is finite-depth then Mk ≃ L(Frk), rk = 1 + 2Iδ−2k(δ − 1), where δ2 = [M1 : M0] and I is the global index.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 5 / 16

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Planar algebra subfactors

Let P = (Pn,±)n≥0 be a subfactor planar algebra. Guionnet-Jones-Shlyakhtenko ’08,’09: Tower of graded algebras Gr0(P) ⊂ Gr1(P) ⊂ · · · . Voiculescu trace τk : Grk(P) → C. GNS completions give tower of II1 factors M0 ⊂ M1 ⊂ · · · , whose planar algebra is P. (Diagrammatic proof of Popa’s reconstruction theorem). If P is finite-depth then Mk ≃ L(Frk), rk = 1 + 2Iδ−2k(δ − 1), where δ2 = [M1 : M0] and I is the global index. Relies on work of K. Dykema on amalgamated free products.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 5 / 16

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Planar algebra subfactors

Let P = (Pn,±)n≥0 be a subfactor planar algebra. Guionnet-Jones-Shlyakhtenko ’08,’09: Tower of graded algebras Gr0(P) ⊂ Gr1(P) ⊂ · · · . Voiculescu trace τk : Grk(P) → C. GNS completions give tower of II1 factors M0 ⊂ M1 ⊂ · · · , whose planar algebra is P. (Diagrammatic proof of Popa’s reconstruction theorem). If P is finite-depth then Mk ≃ L(Frk), rk = 1 + 2Iδ−2k(δ − 1), where δ2 = [M1 : M0] and I is the global index. Relies on work of K. Dykema on amalgamated free products. Kodiyalam-Sunder ’09: Independent proof that Mk are interpolated free group factors.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 5 / 16

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Planar algebra subfactors

Let P = (Pn,±)n≥0 be a subfactor planar algebra. Guionnet-Jones-Shlyakhtenko ’08,’09: Tower of graded algebras Gr0(P) ⊂ Gr1(P) ⊂ · · · . Voiculescu trace τk : Grk(P) → C. GNS completions give tower of II1 factors M0 ⊂ M1 ⊂ · · · , whose planar algebra is P. (Diagrammatic proof of Popa’s reconstruction theorem). If P is finite-depth then Mk ≃ L(Frk), rk = 1 + 2Iδ−2k(δ − 1), where δ2 = [M1 : M0] and I is the global index. Relies on work of K. Dykema on amalgamated free products. Kodiyalam-Sunder ’09: Independent proof that Mk are interpolated free group factors. Connections to random matrix models (GJS+Zinn-Justin ’10).

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 5 / 16

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Graded algebras associated to planar algebras

Let P = (Pn,±)n≥0 be a subfactor planar algebra. For n, k ≥ 0 let Pn,k be a copy of Pn+k, represented by:

k k 2n

x

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 6 / 16

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Graded algebras associated to planar algebras

Let P = (Pn,±)n≥0 be a subfactor planar algebra. For n, k ≥ 0 let Pn,k be a copy of Pn+k, represented by:

k k 2n

x Multiplication: ∧k : Pn,k × Pm,k → Pn+m,k x x ∧k y = y

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 6 / 16

SLIDE 29

Graded algebras associated to planar algebras

Let P = (Pn,±)n≥0 be a subfactor planar algebra. For n, k ≥ 0 let Pn,k be a copy of Pn+k, represented by:

k k 2n

x Multiplication: ∧k : Pn,k × Pm,k → Pn+m,k x x ∧k y = y Voiculescu trace: τk : Pn,k → P0 ≃ C x

TL

τk(x) = δ−k·

Loopless diagrams with 2n boundary points

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 6 / 16

SLIDE 30

Graded algebras associated to planar algebras

Let P = (Pn,±)n≥0 be a subfactor planar algebra. For n, k ≥ 0 let Pn,k be a copy of Pn+k, represented by:

k k 2n

x Multiplication: ∧k : Pn,k × Pm,k → Pn+m,k x x ∧k y = y Voiculescu trace: τk : Pn,k → P0 ≃ C x

TL

τk(x) = δ−k·

Loopless diagrams with 2n boundary points

Graded algebras: (Grk, ∧k, τk), Grk =

n≥0

Pn,k.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 6 / 16

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Key example: Polynomial planar algebra

A = C[X1, X ∗

1 , . . . , Xn, X ∗ n ].

Pm,+ = span{Xi1X ∗

i2 · · · Xi2m−1X ∗ i2m : 1 ≤ i1, . . . , i2m ≤ n}

Pm,− = span{X ∗

i1 Xi2 · · · X ∗ i2m−1Xi2m : 1 ≤ i1, . . . , i2m ≤ n}

Planar tangles act by “contracting indices”.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 7 / 16

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Key example: Polynomial planar algebra

A = C[X1, X ∗

1 , . . . , Xn, X ∗ n ].

Pm,+ = span{Xi1X ∗

i2 · · · Xi2m−1X ∗ i2m : 1 ≤ i1, . . . , i2m ≤ n}

Pm,− = span{X ∗

i1 Xi2 · · · X ∗ i2m−1Xi2m : 1 ≤ i1, . . . , i2m ≤ n}

Planar tangles act by “contracting indices”. ∧0 : Pm × Pk → Pm+k:

Xi1 X ∗

i2

X ∗

i2m

· · · · · · Xj1 X ∗

j2

X ∗

j2k

· · · · · · = Xi1 X ∗

i2

X ∗

i2m

· · · · · · Xj1X ∗

j2

X ∗

j2k

· · · · · ·

Inclusion Gr0 ֒ → A.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 7 / 16

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Key example: Polynomial planar algebra

A = C[X1, X ∗

1 , . . . , Xn, X ∗ n ].

Pm,+ = span{Xi1X ∗

i2 · · · Xi2m−1X ∗ i2m : 1 ≤ i1, . . . , i2m ≤ n}

Pm,− = span{X ∗

i1 Xi2 · · · X ∗ i2m−1Xi2m : 1 ≤ i1, . . . , i2m ≤ n}

Planar tangles act by “contracting indices”. ∧0 : Pm × Pk → Pm+k:

Xi1 X ∗

i2

X ∗

i2m

· · · · · · Xj1 X ∗

j2

X ∗

j2k

· · · · · · = Xi1 X ∗

i2

X ∗

i2m

· · · · · · Xj1X ∗

j2

X ∗

j2k

· · · · · ·

Inclusion Gr0 ֒ → A. τ0: Restriction to Gr0 of the Voiculescu trace on A, i.e. X1, . . . , Xn ∗-free circular random variables.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 7 / 16

SLIDE 34

Key example: Polynomial planar algebra

A = C[X1, X ∗

1 , . . . , Xn, X ∗ n ].

Pm,+ = span{Xi1X ∗

i2 · · · Xi2m−1X ∗ i2m : 1 ≤ i1, . . . , i2m ≤ n}

Pm,− = span{X ∗

i1 Xi2 · · · X ∗ i2m−1Xi2m : 1 ≤ i1, . . . , i2m ≤ n}

Planar tangles act by “contracting indices”. ∧0 : Pm × Pk → Pm+k:

Xi1 X ∗

i2

X ∗

i2m

· · · · · · Xj1 X ∗

j2

X ∗

j2k

· · · · · · = Xi1 X ∗

i2

X ∗

i2m

· · · · · · Xj1X ∗

j2

X ∗

j2k

· · · · · ·

Inclusion Gr0 ֒ → A. τ0: Restriction to Gr0 of the Voiculescu trace on A, i.e. X1, . . . , Xn ∗-free circular random variables. (Grk, τk) ≃ (Gr0 ⊗ Mn(C)⊗k, τ0 ⊗ tr⊗k).

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 7 / 16

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Subfactors associated to planar algebras

Inclusions: Grk ⊂ Grk+1 : x x

→

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 8 / 16

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Subfactors associated to planar algebras

Inclusions: Grk ⊂ Grk+1 : x x

→

Jones projections: ek ∈ Grk+2,

k

ek = δ−1·

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 8 / 16

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Subfactors associated to planar algebras

Inclusions: Grk ⊂ Grk+1 : x x

→

Jones projections: ek ∈ Grk+2,

k

ek = δ−1·

Theorem (Guionnet-Jones-Shlyakhtenko ’08, Popa ’95)

Let P be a subfactor planar algebra. For k ≥ 0 the Voiculescu trace τk is faithful and positive, and the GNS completion Mk is a II1 factor as long as δ > 1. The inclusions Grk ⊂ Grk+1 extend to Mk ⊂ Mk+1, and (Mk, ek) is the Jones tower for the subfactor M0 ⊂ M1, whose planar algebra is P.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 8 / 16

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Popa’s symmetric enveloping algebra

Theorem (Popa ’94)

Let N ⊂ M be an inclusion of II1 factors with [M : N] < ∞. Then there is a (unique up to conjugacy) II1 factor M ⊠eN Mop such that:

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 9 / 16

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Popa’s symmetric enveloping algebra

Theorem (Popa ’94)

Let N ⊂ M be an inclusion of II1 factors with [M : N] < ∞. Then there is a (unique up to conjugacy) II1 factor M ⊠eN Mop such that:

1 There is an anti-automorphism x → xop of M ⊠eN Mop such that

(xop)op = x.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 9 / 16

SLIDE 40

Popa’s symmetric enveloping algebra

Theorem (Popa ’94)

Let N ⊂ M be an inclusion of II1 factors with [M : N] < ∞. Then there is a (unique up to conjugacy) II1 factor M ⊠eN Mop such that:

1 There is an anti-automorphism x → xop of M ⊠eN Mop such that

(xop)op = x.

2 There is an inclusion M ⊗ Mop ֒

→ M ⊠eN Mop such that (x ⊗ yop)op = y ⊗ xop.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 9 / 16

SLIDE 41

Popa’s symmetric enveloping algebra

Theorem (Popa ’94)

Let N ⊂ M be an inclusion of II1 factors with [M : N] < ∞. Then there is a (unique up to conjugacy) II1 factor M ⊠eN Mop such that:

1 There is an anti-automorphism x → xop of M ⊠eN Mop such that

(xop)op = x.

2 There is an inclusion M ⊗ Mop ֒

→ M ⊠eN Mop such that (x ⊗ yop)op = y ⊗ xop.

3 There is a projection eN ∈ M ⊠eN Mop such that eop

N = eN, and eN is

the Jones projection for the inclusions N ⊂ M and Nop ⊂ Mop.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 9 / 16

SLIDE 42

Popa’s symmetric enveloping algebra

Theorem (Popa ’94)

Let N ⊂ M be an inclusion of II1 factors with [M : N] < ∞. Then there is a (unique up to conjugacy) II1 factor M ⊠eN Mop such that:

1 There is an anti-automorphism x → xop of M ⊠eN Mop such that

(xop)op = x.

2 There is an inclusion M ⊗ Mop ֒

→ M ⊠eN Mop such that (x ⊗ yop)op = y ⊗ xop.

3 There is a projection eN ∈ M ⊠eN Mop such that eop

N = eN, and eN is

the Jones projection for the inclusions N ⊂ M and Nop ⊂ Mop.

4 M ⊠eN M is generated by M ⊗ Mop and eN. Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 9 / 16

SLIDE 43

Popa’s symmetric enveloping algebra

Theorem (Popa ’94)

Let N ⊂ M be an inclusion of II1 factors with [M : N] < ∞. Then there is a (unique up to conjugacy) II1 factor M ⊠eN Mop such that:

1 There is an anti-automorphism x → xop of M ⊠eN Mop such that

(xop)op = x.

2 There is an inclusion M ⊗ Mop ֒

→ M ⊠eN Mop such that (x ⊗ yop)op = y ⊗ xop.

3 There is a projection eN ∈ M ⊠eN Mop such that eop

N = eN, and eN is

the Jones projection for the inclusions N ⊂ M and Nop ⊂ Mop.

4 M ⊠eN M is generated by M ⊗ Mop and eN.

M amenable, N ⊂ M finite-depth ⇒ M ⊗ Mop ⊂ M ⊠eN Mop ≃ Ocneanu’s asymptotic inclusion. Index = I, related to Drinfeld double.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 9 / 16

SLIDE 44

Popa’s symmetric enveloping algebra

Theorem (Popa ’94)

Let N ⊂ M be an inclusion of II1 factors with [M : N] < ∞. Then there is a (unique up to conjugacy) II1 factor M ⊠eN Mop such that:

1 There is an anti-automorphism x → xop of M ⊠eN Mop such that

(xop)op = x.

2 There is an inclusion M ⊗ Mop ֒

→ M ⊠eN Mop such that (x ⊗ yop)op = y ⊗ xop.

3 There is a projection eN ∈ M ⊠eN Mop such that eop

N = eN, and eN is

the Jones projection for the inclusions N ⊂ M and Nop ⊂ Mop.

4 M ⊠eN M is generated by M ⊗ Mop and eN.

M amenable, N ⊂ M finite-depth ⇒ M ⊗ Mop ⊂ M ⊠eN Mop ≃ Ocneanu’s asymptotic inclusion. Index = I, related to Drinfeld double. Popa ’94, ’99: M ⊗ Mop ⊂ M ⊠eN Mop encodes a number of important analytic properties of N ⊂ M.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 9 / 16

SLIDE 45

Symmetric enveloping graded algebra

For k, s, t ≥ 0 let Vk,s,t be a copy of P2k+s+t, represented by:

2k 2k 2t 2s

x

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 10 / 16

SLIDE 46

Symmetric enveloping graded algebra

For k, s, t ≥ 0 let Vk,s,t be a copy of P2k+s+t, represented by:

2k 2k 2t 2s

x Multiplication: ∧k : Vk,s,t × Vk,s′,t′ → Vk,s+s′,t+t′ x x ∧k y = y

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 10 / 16

SLIDE 47

Symmetric enveloping graded algebra

For k, s, t ≥ 0 let Vk,s,t be a copy of P2k+s+t, represented by:

2k 2k 2t 2s

x Multiplication: ∧k : Vk,s,t × Vk,s′,t′ → Vk,s+s′,t+t′ x x ∧k y = y Trace: τk ⊠ τk : Vk,s,t → P0 ≃ C. x

TL TL

τk ⊠ τk(x) = δ−2k·

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 10 / 16

SLIDE 48

Symmetric enveloping graded algebra

For k, s, t ≥ 0 let Vk,s,t be a copy of P2k+s+t, represented by:

2k 2k 2t 2s

x Multiplication: ∧k : Vk,s,t × Vk,s′,t′ → Vk,s+s′,t+t′ x x ∧k y = y Trace: τk ⊠ τk : Vk,s,t → P0 ≃ C. x

TL TL

τk ⊠ τk(x) = δ−2k· Graded algebra: (Grk ⊠ Grop

k , ∧k, τk ⊠ τk),

Grk ⊠ Grop

k

=

s,t≥0 Vk,s,t.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 10 / 16

SLIDE 49

Symmetric enveloping inclusion

Anti-automorphism: y ∈ Vk,s,t → yop ∈ Vk,t,s yop = y

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 11 / 16

SLIDE 50

Symmetric enveloping inclusion

Anti-automorphism: y ∈ Vk,s,t → yop ∈ Vk,t,s yop = y Inclusion: Grk ⊗ Grop

k

֒ → Grk ⊠ Grop

k , (x ⊗ yop)op = y ⊗ xop.

x y

x ⊗ yop →

(Polynomial case: Grk ⊠ Grop

k

= Grk ⊗ Grop

k ).

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 11 / 16

SLIDE 51

Symmetric enveloping inclusion

Anti-automorphism: y ∈ Vk,s,t → yop ∈ Vk,t,s yop = y Inclusion: Grk ⊗ Grop

k

֒ → Grk ⊠ Grop

k , (x ⊗ yop)op = y ⊗ xop.

x y

x ⊗ yop →

(Polynomial case: Grk ⊠ Grop

k

= Grk ⊗ Grop

k ).

Jones projection: ek−1 ∈ Vk,0,0.

k − 1 k − 1

ek−1 = δ−1·

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 11 / 16

SLIDE 52

Symmetric enveloping algebra

Theorem (C.-Jones-Shlyakhtenko ’11)

For k ≥ 0, τk ⊠ τk is a faithful, positive state on Grk ⊠ Grop

k . If δ > 1,

k ≥ 1, the GNS completion Mk ⊠ Mop

k

is isomorphic to Popa’s symmetric enveloping algebra Mk ⊠ek−1 Mop

k .

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 12 / 16

SLIDE 53

Symmetric enveloping algebra

Theorem (C.-Jones-Shlyakhtenko ’11)

For k ≥ 0, τk ⊠ τk is a faithful, positive state on Grk ⊠ Grop

k . If δ > 1,

k ≥ 1, the GNS completion Mk ⊠ Mop

k

is isomorphic to Popa’s symmetric enveloping algebra Mk ⊠ek−1 Mop

k .

Proof of positivity uses diagrammatic orthogonalization procedure from Jones-Shlyakhtenko-Walker ’09 (Kodiyalam-Sunder ’09).

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 12 / 16

SLIDE 54

Symmetric enveloping algebra

Theorem (C.-Jones-Shlyakhtenko ’11)

For k ≥ 0, τk ⊠ τk is a faithful, positive state on Grk ⊠ Grop

k . If δ > 1,

k ≥ 1, the GNS completion Mk ⊠ Mop

k

is isomorphic to Popa’s symmetric enveloping algebra Mk ⊠ek−1 Mop

k .

Proof of positivity uses diagrammatic orthogonalization procedure from Jones-Shlyakhtenko-Walker ’09 (Kodiyalam-Sunder ’09). Popa’s characterization of symmetric enveloping algebra implies W ∗(Mk ⊗ Mop

k , ek−1) ≃ Mk ⊠ek−1 Mop k .

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 12 / 16

SLIDE 55

Symmetric enveloping algebra

Theorem (C.-Jones-Shlyakhtenko ’11)

For k ≥ 0, τk ⊠ τk is a faithful, positive state on Grk ⊠ Grop

k . If δ > 1,

k ≥ 1, the GNS completion Mk ⊠ Mop

k

is isomorphic to Popa’s symmetric enveloping algebra Mk ⊠ek−1 Mop

k .

Proof of positivity uses diagrammatic orthogonalization procedure from Jones-Shlyakhtenko-Walker ’09 (Kodiyalam-Sunder ’09). Popa’s characterization of symmetric enveloping algebra implies W ∗(Mk ⊗ Mop

k , ek−1) ≃ Mk ⊠ek−1 Mop k .

Key step is to show: Mk ⊠ Mop

k

= W ∗(Mk ⊗ Mop

k , ek−1).

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 12 / 16

SLIDE 56

Sketch of proof

Step 1: Reduce to showing that if x ∈ Grk ⊠ Grop

k , then for l

sufficiently large we have x

2l 2l 2l 2l

∈ W ∗(Mk+2l ⊗ Mop

k+2l, ek+2l−1)

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 13 / 16

SLIDE 57

Sketch of proof

Step 1: Reduce to showing that if x ∈ Grk ⊠ Grop

k , then for l

sufficiently large we have x

2l 2l 2l 2l

∈ W ∗(Mk+2l ⊗ Mop

k+2l, ek+2l−1)

Step 2: x ∈ Vk,s,t, l = t + k, x

2l 2l 2l 2l

= δ−(l+t)·

x

2l 2l k k 2k l t 2l

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 13 / 16

SLIDE 58

Sketch of proof

Step 1: Reduce to showing that if x ∈ Grk ⊠ Grop

k , then for l

sufficiently large we have x

2l 2l 2l 2l

∈ W ∗(Mk+2l ⊗ Mop

k+2l, ek+2l−1)

Step 2: x ∈ Vk,s,t, l = t + k, x

2l 2l 2l 2l

= δ−(l+t)·

x

2l 2l k k 2k l t 2l Mk+2l ⊗ Mop

k+2l

∈ Mk+2l ⊗ Mop

k+2l

∈

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 13 / 16

SLIDE 59

Sketch of proof

Step 1: Reduce to showing that if x ∈ Grk ⊠ Grop

k , then for l

sufficiently large we have x

2l 2l 2l 2l

∈ W ∗(Mk+2l ⊗ Mop

k+2l, ek+2l−1)

Step 2: x ∈ Vk,s,t, l = t + k, x

2l 2l 2l 2l

= δ−(l+t)·

x

2l 2l k k 2k l t 2l ∈

i

TL(k + 2l) = Alg

Stephen Curran (UCLA)

Symmetric enveloping algebra May 26, 2011 13 / 16

SLIDE 60

On the symmetric enveloping inclusion

Theorem (Ocneanu ’88, Popa ’94)

As a M0 ⊗ Mop

0 -bimodule,

L2(M0 ⊠ M0) ≃

v∈Γ+

Xv ⊗ Xv, where Γ is the principal graph of the inclusion M0 ⊂ M1.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 14 / 16

SLIDE 61

On the symmetric enveloping inclusion

Theorem (Ocneanu ’88, Popa ’94)

As a M0 ⊗ Mop

0 -bimodule,

L2(M0 ⊠ M0) ≃

v∈Γ+

Xv ⊗ Xv, where Γ is the principal graph of the inclusion M0 ⊂ M1. In particular, [M0 ⊠ M0 : M0 ⊗ Mop

0 ] =

v∈Γ+

dimM0(Xv)2 = I.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 14 / 16

SLIDE 62

On the symmetric enveloping inclusion

Theorem (Ocneanu ’88, Popa ’94)

As a M0 ⊗ Mop

0 -bimodule,

L2(M0 ⊠ M0) ≃

v∈Γ+

Xv ⊗ Xv, where Γ is the principal graph of the inclusion M0 ⊂ M1. In particular, [M0 ⊠ M0 : M0 ⊗ Mop

0 ] =

v∈Γ+

dimM0(Xv)2 = I. Can give a diagrammatic description of the decomposition above.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 14 / 16

SLIDE 63

On the symmetric enveloping inclusion

Theorem (Ocneanu ’88, Popa ’94)

As a M0 ⊗ Mop

0 -bimodule,

L2(M0 ⊠ M0) ≃

v∈Γ+

Xv ⊗ Xv, where Γ is the principal graph of the inclusion M0 ⊂ M1. In particular, [M0 ⊠ M0 : M0 ⊗ Mop

0 ] =

v∈Γ+

dimM0(Xv)2 = I. Can give a diagrammatic description of the decomposition above. Work in progress: Use this description to compute the planar algebra

f M0 ⊗ Mop

⊂ M0 ⊠ M0 when P is finite-depth (computed in terms

f connections by Evans-Kawahigashi ’95).

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 14 / 16

SLIDE 64

Derivations and free entropy dimension

Theorem (Guionnet-Jones-Shylyakhtenko ’09)

If P is finite-depth, then M0 ≃ LFr0 where r0 = 1 + 2I(δ − 1).

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 15 / 16

SLIDE 65

Derivations and free entropy dimension

Theorem (Guionnet-Jones-Shylyakhtenko ’09)

If P is finite-depth, then M0 ≃ LFr0 where r0 = 1 + 2I(δ − 1).

Goal: Try to understand the formula for r0 by considering derivations on M0.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 15 / 16

SLIDE 66

Derivations and free entropy dimension

Theorem (Guionnet-Jones-Shylyakhtenko ’09)

If P is finite-depth, then M0 ≃ LFr0 where r0 = 1 + 2I(δ − 1).

Goal: Try to understand the formula for r0 by considering derivations on M0. Disclaimer: This is purely heuristic!

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 15 / 16

SLIDE 67

Derivations and free entropy dimension

Theorem (Guionnet-Jones-Shylyakhtenko ’09)

If P is finite-depth, then M0 ≃ LFr0 where r0 = 1 + 2I(δ − 1).

Goal: Try to understand the formula for r0 by considering derivations on M0. Disclaimer: This is purely heuristic! M = W ∗(X1, . . . , Xn). A = CX1, . . . , Xn. V = {(δ(X1), . . . , δ(Xn) : δ ∈ Derc(A, A ⊗ Aop)} ⊂ L2(M ⊗ Mop)n.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 15 / 16

SLIDE 68

Derivations and free entropy dimension

Theorem (Guionnet-Jones-Shylyakhtenko ’09)

If P is finite-depth, then M0 ≃ LFr0 where r0 = 1 + 2I(δ − 1).

Goal: Try to understand the formula for r0 by considering derivations on M0. Disclaimer: This is purely heuristic! M = W ∗(X1, . . . , Xn). A = CX1, . . . , Xn. V = {(δ(X1), . . . , δ(Xn) : δ ∈ Derc(A, A ⊗ Aop)} ⊂ L2(M ⊗ Mop)n. Observation: If X1, . . . , Xn are free semicircular then W∗(X1, . . . , Xn) ≃ LFn and dimM⊗Mop V

L2(M⊗Mop)n

= n.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 15 / 16

SLIDE 69

Derivations and free entropy dimension

Theorem (Guionnet-Jones-Shylyakhtenko ’09)

If P is finite-depth, then M0 ≃ LFr0 where r0 = 1 + 2I(δ − 1).

Goal: Try to understand the formula for r0 by considering derivations on M0. Disclaimer: This is purely heuristic! M = W ∗(X1, . . . , Xn). A = CX1, . . . , Xn. V = {(δ(X1), . . . , δ(Xn) : δ ∈ Derc(A, A ⊗ Aop)} ⊂ L2(M ⊗ Mop)n. Observation: If X1, . . . , Xn are free semicircular then W∗(X1, . . . , Xn) ≃ LFn and dimM⊗Mop V

L2(M⊗Mop)n

= n. Idea: For “nice” generators X1, . . . , Xn of LFt, we should have dimM⊗Mop V

L2(M⊗Mop)n

= t.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 15 / 16

SLIDE 70

Derivations and free entropy dimension

Theorem (Guionnet-Jones-Shylyakhtenko ’09)

If P is finite-depth, then M0 ≃ LFr0 where r0 = 1 + 2I(δ − 1).

Goal: Try to understand the formula for r0 by considering derivations on M0. Disclaimer: This is purely heuristic! M = W ∗(X1, . . . , Xn). A = CX1, . . . , Xn. V = {(δ(X1), . . . , δ(Xn) : δ ∈ Derc(A, A ⊗ Aop)} ⊂ L2(M ⊗ Mop)n. Observation: If X1, . . . , Xn are free semicircular then W∗(X1, . . . , Xn) ≃ LFn and dimM⊗Mop V

L2(M⊗Mop)n

= n. Idea: For “nice” generators X1, . . . , Xn of LFt, we should have dimM⊗Mop V

L2(M⊗Mop)n

= t. Biane-Capitaine-Guionnet ’03 + Connes-Shlyakhtenko ’05, Shlyakhtenko ’09, Dabrowski ’10: Estimates on δ0(X1, . . . , Xn) in terms of MvN-dimensions of related spaces of derivations.

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 15 / 16

SLIDE 71

Diagrammatic derivations on Gr0(P)

s + t + 1 = 2n, Φ(s, t) a copy of Pn represented as:

t s

Q Right Gr0 ⊗ Grop

0 -module: Φ = Φ(s, t).

Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 16 / 16

SLIDE 72

Diagrammatic derivations on Gr0(P)

s + t + 1 = 2n, Φ(s, t) a copy of Pn represented as:

t s

Q Right Gr0 ⊗ Grop

0 -module: Φ = Φ(s, t).

Q ∈ Φ(s, t) ⇒ δQ ∈ Der(Gr0, Gr0 ⊗ Grop

0 ),

x Q k δQ(x) =

0≤k<2n

k+t even

EM0⊗Mop

Stephen Curran (UCLA)

Symmetric enveloping algebra May 26, 2011 16 / 16