Commutants of multifusion categories

Transcription

1 Commutants of multifusion categories David Pennes, OSU joint with André Henriques AMS Special Session on Fusion categories and applications April 1, 2017

2 Categorical analogies Tensor categories categorif algebras. algebra A tensor categor C finite dimensional algebra fusion categor center Z(A) Drinfel d center Z(C) commutant Z B (A) of A in B commutant Z D (C) of C in D B(H) Bim(R), all bimodules commutant A := Z B(H) (A) commutant C := Z Bim(R) (C) von Neumann algebra A = A bicommutant categor C = C Bicommutant categories categorif von Neumann algebras.

3 Categorifing basic theorems In previous work with Henriques, we proved the categorified finite dimensional bicommutant theorem. Theorem [HP15] Suppose C is a unitar fusion categor embedded in Bim(R), where R is a non tpe I factor. Then C = C Vec Hilb = Hilb(C). Toda, we will prove the categorification of: Two Morita equivalent finite dimensional von Neumann algebras embedded in B(H) have isomorphic commutants.

4 Unitar multifusion categories Definition A k k unitar multifusion categor is a rigid C*-tensor categor C satisfing: C is idempotent complete, C has finitel man isomorphism classes of simple objects, 1 C = k i=1 1 i where each 1 i is simple, and C is indecomposable. Proposition Ever unitar k k multifusion categor has a full faithful tensor embedding C Bim(R k ) which is dimension preserving. The proof uses a modification of Ocneanu compactness [JS97].

5 Graphical calculus Fi a finite set Irr(C) of representatives of irreducibles. Shaded regions denote irreducible summands of 1 C. Morphisms f : are represented b coupons. For all simple C i,j, C j,k, and, C k,i, Hom(1, ) is a finite dimensional Hilbert space with inner product f, g = g f. Choose dual bases: e i Hom(1, ) and e i Hom(1, ) We represent the canonical element b colored nodes := d d d α e α e α The canonical element is independent of choice of basis.

6 Important relations = d d d 1 N, (Bigon 1) = d d d 1 (Bigon 2) d = d d (Fusion) Irr(C ik ) v v Irr(C il ) w v = u Irr(C w jk ) w u u w (I=H) We ll use Snder convention and ignore all scalars.

7 Commutant C of C in Bim(R k ) The commutant C Bim(R k ) of C Bim(R k ) has: Objects are pairs (X, e X ) where X Bim(R k ), and e X is a unitar half braiding with C e X,c = X c : X c c X These half braidings must satisf compatibilit conditions. Morphisms f : (X, e X ) (Y, e Y ) are bimodule maps f : X Y which commute with the half braidings: Y f = f Y X c X c C is a tensor categor, but it is usuall not braided.

8 Describing C for unitar multifusion Suppose (X, e X ) C Bim(R k ). Write X = (X i,j ) where X i,j is an R i R j bimodule. Eas facts about (X, e X ) C 1. X 1 j = 1j X 1 j implies X i,j = 0 for i j. 2. Writing X = k i=1 X i, we can write e X as a famil of natural isomorphisms (e i X ) given on c i,j C i,j b c i,j X j = e i X Hom Ri R j (X i Ri c i,j c i,j Rj X j ) X i c i,j 3. We have a projection functor P j : C C j b (X, e X ) (X j, e j X ).

9 Induction functor Bim(R) C We have a wa to construct lots of objects in C. We alwas use the shading = R 1. Φ(Λ) := Φ : Bim(R 1 ) C Φ(Λ) = (Φ(Λ), e Φ(Λ) ). j=1,...,k c Irr(C j,1 ) c Λ c Bim(R k ) e Φ(Λ),a := i,j {1,...,k} =R i =R j b Irr(C j,1 ) c Irr(C i,1 ) d 1 a a b Λ b c Λ c a ; a C i,j Proposition The functor Φ : Bim(R 1 ) C is dominant.

10 A canonical projector For (X 1, e X1 ) C 1, we have a canonical projector in End C (Φ(X 1 )): p X1 : = 1 D = 1 D j {1,...,k} =R j j {1,...,k} =R j a Irr(C 1 ), Irr(C j,1 ), Irr(C j,1 ) da d d X 1 a a X 1 X 1 X j X 1

11 Equivalence We have functors Bim(R k ) C P 1 C 1 Bim(R 1 ) Bim(R 1 ) C 1 Φ C Bim(R k ) We get another functor pφ : C 1 C b appling Φ and then appling the canonical projector. Theorem The functors P 1 and pφ witness an equivalence of categories Bim(R 1 ) C 1 = C Bim(R k ). Sketch of one direction. We get a natural isomorphism u : pφ P 1 id where u X : p X Φ(X 1 ) X is given b u X = 1 D j {1,...,k} =R j Irr(C j,1 ) d X j X 1.

12 The main corollar We can now prove our main result as a corollar. Corollar If C 1 Bim(R 1 ) and C 2 Bim(R 2 ) are two Morita equivalent unitar fusion categories, then C 1 Bim(R 1) = C 2 Bim(R 2). Proof. Let M Bim(R 1, R 2 ) be an equivalence unitar C 1 C 2 bimodule categor. We can form a 2 2 unitar multifusion categor b ( ) C1 M C = M Bim(R C 1 R 2 ). 2 Now we appl the previous theorem twice: C 1 Bim(R 1 ) = C Bim(R 1 R 2 ) = C 2 Bim(R 2 ).

13 Thank ou for listening! Slides available at: https: //people.math.osu.edu/pennes.2/pennesams2017.pdf Previous article Bicommutant categories from fusion categories with André Henriques available at: New article Commutants of multifusion categories with André Henriques coming soon!