Representation theory and (co)homology for subfactors, -lattices and - PowerPoint PPT Presentation

Representation theory and (co)homology for subfactors, λ -lattices and C ∗ -tensor categories Abel Symposium, 7-11 August 2015 Stefaan Vaes ∗ (joint work with Sorin Popa and Dimitri Shlyakhtenko) ∗ Supported by ERC Consolidator Grant 614195 1/12

A very short introduction to subfactors Goal: representation theory and (co)homology for the standard invariant of subfactors. Let N ⊂ M be a finite index subfactor. ◮ Jones projection e N : L 2 ( M ) → L 2 ( N ). ◮ Basic construction M 1 = � M , e N � , with canonical tracial state τ . ◮ Jones tower N ⊂ M ⊂ M 1 ⊂ M 2 ⊂ · · · . ◮ C is the category of all M -bimodules that appear in some M L 2 ( M n ) M . ◮ Extremality : these bimodules have equal left and right dimension. ◮ Standard invariant : the λ -lattice of multimatrix algebras M ′ k ∩ M n , k ≤ n . 2/12

Popa’s symmetric enveloping algebra Extremal finite index subfactor N ⊂ M a crossed product like inclusion T ⊂ S . ◮ T = M ⊗ M op . ◮ S = M ⊠ e N M op is the unique II 1 factor • generated by commuting copies of M and M op , • and a projection e N , • that is the Jones projection for both N ⊂ M and N op ⊂ M op . ◮ We have T L 2 ( S ) T ∼ � = M ⊗ M op ( H α ⊗ H α ) M ⊗ M op . α ∈ Irr C Recall : C is the category of M -bimodules appearing in some L 2 ( M n ). Terminology : the SE-inclusion of N ⊂ M . 3/12

SE-correspondences Definition (Popa-V, 2014) An SE-correspondence of N ⊂ M is an S -bimodule that is generated by T -central vectors. Here : T ⊂ S is the SE-inclusion of N ⊂ M . ◮ Trivial SE-correspondence : L 2 ( S ). ◮ Regular (coarse) SE-correspondence : L 2 ( S ) ⊗ T L 2 ( S ). ◮ Weak containment, amenability, Haagerup property, property (T) now have “obvious” definitions. Questions : Does all this only depend on the standard invariant ? What about basic examples ? 4/12

Positive definite functions : cp SE-multipliers Analogy : T ⊂ S = T ⋊ Γ. A one-to-one correspondence between ◮ positive definite functions ϕ : Γ → C , ◮ and normal, completely positive T -bimodular maps ψ : S → S . Explicitly : ψ ( au g ) = ϕ ( g ) au g . Definition (Popa-V, 2014) A cp SE-multiplier of N ⊂ M is a normal, completely positive T -bimodule map ψ : S → S . ◮ Similarly : cb SE-multipliers , CMAP, weak amenability. ◮ Since L 2 ( S ) ∼ = � α ∈ Irr C ( H α ⊗ H α ) as T -bimodules, a T -bimodular map ψ : S → S must be given by multiplication with a scalar ϕ ( α ) on each H α ⊗ H α . ◮ Popa-V : an intrinsic description of positive definiteness, on arbitrary rigid C ∗ -tensor categories. 5/12

Universal C ∗ -algebra of a rigid C ∗ -tensor category Fusion ∗ -algebra C [ C ] with Irr( C ) as vector space basis and product given by the fusion rules. ◮ Every cp-multiplier ψ : Irr( C ) → C defines a positive functional ω ψ : C [ C ] → C : ω ψ ( α ) = d ( α ) ψ ( α ). ◮ Not every positive functional on C [ C ] arises like this. ◮ We have singled out the class of admissible representations of C [ C ]. ◮ We can define the universal C ∗ -algebra C u ( C ). Obs : the ∗ -algebra C [ C ] need not have a universal enveloping C ∗ -algebra. Theorem (Popa-V, 2014) There is a natural bijection between SE-correspondences of N ⊂ M and representations of the C ∗ -algebra C u ( C ). 6/12

Examples Temperley-Lieb-Jones ◮ The smallest possible λ -lattice: generated by the Jones projections of a subfactor of index λ − 1 . ◮ As a tensor category C = Rep(PSU q (2)) with q + 1 / q = λ − 1 / 2 . ◮ Popa-V : Haagerup property and CMAP hold ; irreducible representations labeled by [0 , λ − 1 ]. Examples with property (T) ◮ Popa-V : the tensor category Rep( SU q (3)) with 0 < q < 1 has property (T). ◮ Gives rise to the first subfactors with a property (T) standard invariant, not constructed from property (T) groups. Behind all this : compact quantum groups and the results of De Commer - Freslon - Yamashita (2013) and Arano (2014). 7/12

Homology and cohomology SE-inclusions T ⊂ S are examples of irreducible quasi-regular inclusions of II 1 factors. ◮ The normalizer N S ( T ) consists of all unitaries u ∈ S with uTu ∗ = T . ◮ The quasi-normalizer QN S ( T ) consists of all x ∈ S such that TxT has finite index as a T -bimodule. Note that : T ⊂ S is quasi-regular, i.e. QN S ( T ) ′′ = S . Rest of the talk (joint work with S. Popa and D. Shlyakhtenko) ◮ (Co)homology for irreducible, quasi-regular inclusions. ◮ In particular, SE-inclusions. Thus: for rigid C ∗ -tensor categories. ◮ L 2 -Betti numbers in all these cases. 8/12

Cohomology for quasi-regular inclusions Definition (PSV, 2015) Let T ⊂ S be an irreducible quasi-regular inclusion and put S = QN S ( T ). For an S -bimodule H , define H n ( T ⊂ S , H ) as the cohomology of d 0 d 1 d 2 H T → Mor T ( S , H ) → Mor T ( S ⊗ T S , H ) → · · · where e.g. ( d 1 c )( x ⊗ y ) = x · c ( y ) − c ( xy ) + c ( x ) · y . ◮ Regular representation : the S -bimodule L 2 ( S ) ⊗ T L 2 ( S ). ◮ Define M := End S − S ( L 2 ( S ) ⊗ T L 2 ( S )) with state µ given by 1 ⊗ 1. ◮ The state µ on M is faithful. It is a trace iff the T -bimodules inside L 2 ( S ) have equal left and right dimension. Definition - for unimodular inclusions (PSV, 2015) β (2) n ( T ⊂ S ) = dim M H n ( T ⊂ S , L 2 ( S ) ⊗ T L 2 ( S )). 9/12

Further remarks ◮ A slight variant works for arbitrary quasi-regular inclusions T ⊂ S . In particular, for Cartan inclusions : we recover Gaboriau’s L 2 -Betti numbers of the associated equivalence relation. ◮ The SE-inclusion of an extremal subfactor is unimodular, and we can thus consider its L 2 -Betti numbers. ◮ Same remark for the SE-inclusion of any tensor category C of finite index M -bimodules with equal left and right dimension. 10/12

Homology for quasi-regular inclusions ◮ Similar bar complex to define homology of a quasi-regular inclusion. ◮ But : how to compute (co)homology for SE-inclusions T ⊂ S ? Tool : Ocneanu’s tube algebra A with augmentation ǫ : A → C . Note : natural projection p 0 ∈ A with p 0 A p 0 = C [ C ]. Theorem (PSV, 2015) Let C be a tensor category of M -bimodules, with Ocneanu’s tube algebra A with augmentation ǫ : A → C , and SE-inclusion T ⊂ S . Then, the Hochschild homology of an A -module K is canonically isomorphic to H n ( T ⊂ S , H ) for the associated S -bimodule H . Intrinsic definition of β (2) n ( C ) for any rigid C ∗ -tensor category C . 11/12

Computations and further results ◮ For Temperley-Lieb-Jones, all L 2 -Betti numbers vanish. ◮ Expected formulae for free products, direct products : non-vanishing β (2) for Fuss-Catalan. 1 ◮ One-cocycles can be exponentiated into cp multipliers. Characterizations of the Haagerup property, property (T), etc. 12/12