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

On the symmetric enveloping algebra of planar algebra subfactors (Joint work with V. Jones and D. Shlyakhtenko) Stephen Curran UCLA Workshop on II 1 factors Institut Henri Poincar´ e May 26, 2011 Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 1 / 16

Planar algebras A (subfactor) planar algebra is a sequence of finite dimensional vector spaces ( P n , ± ) n ≥ 0 with an action of planar tangles . Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

Planar algebras A (subfactor) planar algebra is a sequence of finite dimensional vector spaces ( P n , ± ) n ≥ 0 with an action of planar tangles . D 1 T = D 2 Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

Planar algebras A (subfactor) planar algebra is a sequence of finite dimensional vector spaces ( P n , ± ) n ≥ 0 with an action of planar tangles . D 1 T = D 2 Z T : P 3 , − ⊗ P 2 , + → P 3 , + Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

Planar algebras A (subfactor) planar algebra is a sequence of finite dimensional vector spaces ( P n , ± ) n ≥ 0 with an action of planar tangles . D 1 T = S = D 2 Z T : P 3 , − ⊗ P 2 , + → P 3 , + Z S : P 0 → P 2 , + Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

Planar algebras A (subfactor) planar algebra is a sequence of finite dimensional vector spaces ( P n , ± ) n ≥ 0 with an action of planar tangles . D 1 T ◦ 2 S = S = Z T : P 3 , − ⊗ P 2 , + → P 3 , + Z S : P 0 → P 2 , + Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

Planar algebras A (subfactor) planar algebra is a sequence of finite dimensional vector spaces ( P n , ± ) n ≥ 0 with an action of planar tangles . D 1 T ◦ 2 S = S = Z T : P 3 , − ⊗ P 2 , + → P 3 , + Z S : P 0 → P 2 , + Z T ◦ 2 S = Z T ◦ 2 Z S Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

Planar algebras A (subfactor) planar algebra is a sequence of finite dimensional vector spaces ( P n , ± ) n ≥ 0 with an action of planar tangles . D 1 T = S = D 2 Z T : P 3 , − ⊗ P 2 , + → P 3 , + Z S : P 0 → P 2 , + Z T ◦ 2 S = Z T ◦ 2 Z S Further conditions: P 0 = C , ∗ -structure, positivity, sphericality. Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

Planar algebras A (subfactor) planar algebra is a sequence of finite dimensional vector spaces ( P n , ± ) n ≥ 0 with an action of planar tangles . D 1 T = S = D 2 Z T : P 3 , − ⊗ P 2 , + → P 3 , + Z S : P 0 → P 2 , + Z T ◦ 2 S = Z T ◦ 2 Z S Further conditions: P 0 = C , ∗ -structure, positivity, sphericality. Follows that there is δ ∈ { 2 cos( π/ n ) : n ≥ 3 } ∪ [2 , ∞ ] s.t. Z T ′ = δ · Z T Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

Planar algebras A (subfactor) planar algebra is a sequence of finite dimensional vector spaces ( P n , ± ) n ≥ 0 with an action of planar tangles . D 1 T ′ = S = D 2 Z T : P 3 , − ⊗ P 2 , + → P 3 , + Z S : P 0 → P 2 , + Z T ◦ 2 S = Z T ◦ 2 Z S Further conditions: P 0 = C , ∗ -structure, positivity, sphericality. Follows that there is δ ∈ { 2 cos( π/ n ) : n ≥ 3 } ∪ [2 , ∞ ] s.t. Z T ′ = δ · Z T Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 2 / 16

The polynomial planar algebra P k , ± ⊂ C � X 1 , X ∗ 1 , . . . , X n , X ∗ n P k , + = span { X i 1 X ∗ j 1 · · · X i k X ∗ j k } P k , − = span { X ∗ i 1 X j 1 · · · X ∗ i k X j k } Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 3 / 16

The polynomial planar algebra P k , ± ⊂ C � X 1 , X ∗ 1 , . . . , X n , X ∗ n P k , + = span { X i 1 X ∗ j 1 · · · X i k X ∗ j k } P k , − = span { X ∗ i 1 X j 1 · · · X ∗ i k X j k } T = Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 3 / 16

The polynomial planar algebra P k , ± ⊂ C � X 1 , X ∗ 1 , . . . , X n , X ∗ n P k , + = span { X i 1 X ∗ j 1 · · · X i k X ∗ j k } P k , − = span { X ∗ i 1 X j 1 · · · X ∗ i k X j k } T = Z T ( X ∗ i 1 X j 1 X ∗ i 2 X j 2 X ∗ i 3 X j 3 ⊗ X k 1 X ∗ l 1 X k 2 X ∗ l 2 ) = Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 3 / 16

The polynomial planar algebra P k , ± ⊂ C � X 1 , X ∗ 1 , . . . , X n , X ∗ n P k , + = span { X i 1 X ∗ j 1 · · · X i k X ∗ j k } P k , − = span { X ∗ i 1 X j 1 · · · X ∗ i k X j k } j 2 i 3 j 3 i 2 i 1 j 1 T = l 2 k 1 k 2 l 1 Z T ( X ∗ i 1 X j 1 X ∗ i 2 X j 2 X ∗ i 3 X j 3 ⊗ X k 1 X ∗ l 1 X k 2 X ∗ l 2 ) = Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 3 / 16

The polynomial planar algebra P k , ± ⊂ C � X 1 , X ∗ 1 , . . . , X n , X ∗ n P k , + = span { X i 1 X ∗ j 1 · · · X i k X ∗ j k } P k , − = span { X ∗ i 1 X j 1 · · · X ∗ i k X j k } i 3 j 2 k 1 j 2 i 3 j 3 i 2 i 1 j 1 T = l 2 k 1 k 2 l 1 i 2 l 1 k 2 Z T ( X ∗ i 1 X j 1 X ∗ i 2 X j 2 X ∗ i 3 X j 3 ⊗ X k 1 X ∗ l 1 X k 2 X ∗ l 2 ) = δ j 3 i 1 δ j 1 l 2 X j 2 X ∗ i 3 X k 1 X ∗ l 1 X k 2 X ∗ i 2 Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 3 / 16

The polynomial planar algebra P k , ± ⊂ C � X 1 , X ∗ 1 , . . . , X n , X ∗ n P k , + = span { X i 1 X ∗ j 1 · · · X i k X ∗ j k } P k , − = span { X ∗ i 1 X j 1 · · · X ∗ i k X j k } i 3 j 2 k 1 j 2 i 3 j 3 i 2 i 1 j 1 T = l 2 k 1 k 2 l 1 i 2 l 1 k 2 Z T ( X ∗ i 1 X j 1 X ∗ i 2 X j 2 X ∗ i 3 X j 3 ⊗ X k 1 X ∗ l 1 X k 2 X ∗ l 2 ) = δ j 3 i 1 δ j 1 l 2 X j 2 X ∗ i 3 X k 1 X ∗ l 1 X k 2 X ∗ i 2 Planar algebra of modulus δ = n . Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 3 / 16

Planar algebras and subfactors Theorem (Jones ’99) The standard invariant of any finite-index inclusion of II 1 factors N ⊂ M has a planar algebra structure (with δ 2 = [ M : N ] ). Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 4 / 16

Planar algebras and subfactors Theorem (Jones ’99) The standard invariant of any finite-index inclusion of II 1 factors N ⊂ M has a planar algebra structure (with δ 2 = [ M : N ] ). Theorem (Popa ’95) Any planar algebra ( λ -lattice) is the standard invariant of a finite-index inclusion of II 1 factors. Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 4 / 16

Planar algebra subfactors Let P = ( P n , ± ) n ≥ 0 be a subfactor planar algebra. Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 5 / 16

Planar algebra subfactors Let P = ( P n , ± ) n ≥ 0 be a subfactor planar algebra. Guionnet-Jones-Shlyakhtenko ’08,’09: Tower of graded algebras Gr 0 ( P ) ⊂ Gr 1 ( P ) ⊂ · · · . Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 5 / 16

Planar algebra subfactors Let P = ( P n , ± ) n ≥ 0 be a subfactor planar algebra. Guionnet-Jones-Shlyakhtenko ’08,’09: Tower of graded algebras Gr 0 ( P ) ⊂ Gr 1 ( P ) ⊂ · · · . Voiculescu trace τ k : Gr k ( P ) → C . Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 5 / 16

Planar algebra subfactors Let P = ( P n , ± ) n ≥ 0 be a subfactor planar algebra. Guionnet-Jones-Shlyakhtenko ’08 ,’09: Tower of graded algebras Gr 0 ( P ) ⊂ Gr 1 ( P ) ⊂ · · · . Voiculescu trace τ k : Gr k ( P ) → C . GNS completions give tower of II 1 factors M 0 ⊂ M 1 ⊂ · · · , whose planar algebra is P . (Diagrammatic proof of Popa’s reconstruction theorem). Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 5 / 16

Planar algebra subfactors Let P = ( P n , ± ) n ≥ 0 be a subfactor planar algebra. Guionnet-Jones-Shlyakhtenko ’08, ’09 : Tower of graded algebras Gr 0 ( P ) ⊂ Gr 1 ( P ) ⊂ · · · . Voiculescu trace τ k : Gr k ( P ) → C . GNS completions give tower of II 1 factors M 0 ⊂ M 1 ⊂ · · · , whose planar algebra is P . (Diagrammatic proof of Popa’s reconstruction theorem). If P is finite-depth then M k ≃ L ( F r k ), r k = 1 + 2 I δ − 2 k ( δ − 1) , where δ 2 = [ M 1 : M 0 ] and I is the global index . Stephen Curran (UCLA) Symmetric enveloping algebra May 26, 2011 5 / 16

Planar algebra subfactors Let P = ( P n , ± ) n ≥ 0 be a subfactor planar algebra. Guionnet-Jones-Shlyakhtenko ’08,’09: Tower of graded algebras Gr 0 ( P ) ⊂ Gr 1 ( P ) ⊂ · · · . Voiculescu trace τ k : Gr k ( P ) → C . GNS completions give tower of II 1 factors M 0 ⊂ M 1 ⊂ · · · , whose planar algebra is P . (Diagrammatic proof of Popa’s reconstruction theorem). If P is finite-depth then M k ≃ L ( F r k ), r k = 1 + 2 I δ − 2 k ( δ − 1) , where δ 2 = [ M 1 : M 0 ] 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