Motivation Modular tensor categories Small quasi-quantum groups and modularization Non-semisimple modular tensor categories from quasi-quantum groups Tobias Ohrmann Leibniz University Hannover August 06, 2019
Motivation Modular tensor categories Small quasi-quantum groups and modularization Motivation Modular tensor categories (MTCs) are central objects in
Motivation Modular tensor categories Small quasi-quantum groups and modularization Motivation Modular tensor categories (MTCs) are central objects in 1 2d conformal field theory (2d CFT): chiral half is encoded by representation category of underlying vertex operator algebra (VOA)
Motivation Modular tensor categories Small quasi-quantum groups and modularization Motivation Modular tensor categories (MTCs) are central objects in 1 2d conformal field theory (2d CFT): chiral half is encoded by representation category of underlying vertex operator algebra (VOA) 2 low-dimensional topology : Modular tensor categories yield invariants of oriented closed 3-manifolds more generally: 3d topological field theories [Reshitikin,Turaev][Turaev,Viro]
Motivation Modular tensor categories Small quasi-quantum groups and modularization Motivation Modular tensor categories (MTCs) are central objects in 1 2d conformal field theory (2d CFT): chiral half is encoded by representation category of underlying vertex operator algebra (VOA) 2 low-dimensional topology : Modular tensor categories yield invariants of oriented closed 3-manifolds more generally: 3d topological field theories [Reshitikin,Turaev][Turaev,Viro] 3 FFRS-construction : chiral CFT + special symmetric Frobenius algebra in corresponding MTC ⇒ full CFT
Motivation Modular tensor categories Small quasi-quantum groups and modularization Motivation Above results are 1 proven only in the rational case :
Motivation Modular tensor categories Small quasi-quantum groups and modularization Motivation Above results are 1 proven only in the rational case : Def: VOA V rational : ⇔ V -Rep is finite + semisimple
Motivation Modular tensor categories Small quasi-quantum groups and modularization Motivation Above results are 1 proven only in the rational case : Def: VOA V rational : ⇔ V -Rep is finite + semisimple Huang’04: V -Rep is rational MTC
Motivation Modular tensor categories Small quasi-quantum groups and modularization Motivation Above results are 1 proven only in the rational case : Def: VOA V rational : ⇔ V -Rep is finite + semisimple Huang’04: V -Rep is rational MTC Zhu’96: characters of V are modular invariant
Motivation Modular tensor categories Small quasi-quantum groups and modularization Motivation Above results are 1 proven only in the rational case : Def: VOA V rational : ⇔ V -Rep is finite + semisimple Huang’04: V -Rep is rational MTC Zhu’96: characters of V are modular invariant 2 believed/partly shown to have analogues in non-rational case
Motivation Modular tensor categories Small quasi-quantum groups and modularization Motivation Above results are 1 proven only in the rational case : Def: VOA V rational : ⇔ V -Rep is finite + semisimple Huang’04: V -Rep is rational MTC Zhu’96: characters of V are modular invariant 2 believed/partly shown to have analogues in non-rational case In the talk : keep finiteness, drop semisimplicity
Motivation Modular tensor categories Small quasi-quantum groups and modularization Example: W ( p )-algebras [Kausch’91][Flohr’96][PRZ’06][FGST’06],...: Family of logarithmic CFTs associated to Virasoro ( p , 1)-minimal models: W ( p )-algebras
Motivation Modular tensor categories Small quasi-quantum groups and modularization Example: W ( p )-algebras [Kausch’91][Flohr’96][PRZ’06][FGST’06],...: Family of logarithmic CFTs associated to Virasoro ( p , 1)-minimal models: W ( p )-algebras General construction : Input data: finite dim. simple complex simply laced Lie algebra g , 2 p th root of unit q [Feigin,Tipunin’10]: General approach to construct non-semisimple vertex algebra W g ( p ) from this data
Motivation Modular tensor categories Small quasi-quantum groups and modularization Example: W ( p )-algebras [Kausch’91][Flohr’96][PRZ’06][FGST’06],...: Family of logarithmic CFTs associated to Virasoro ( p , 1)-minimal models: W ( p )-algebras General construction : Input data: finite dim. simple complex simply laced Lie algebra g , 2 p th root of unit q [Feigin,Tipunin’10]: General approach to construct non-semisimple vertex algebra W g ( p ) from this data [Feigin,Tipunin’10][Adamovic,Milas’14],...: W g ( p )-mod ∼ Conjecture: = u -mod (1) as modular tensor categories for some finite dim. factorizable ribbon quasi-Hopf algebra u .
Motivation Modular tensor categories Small quasi-quantum groups and modularization Definition: Modular tensor category Definition Let C be a finite abelian k -linear tensor category. If C has
Motivation Modular tensor categories Small quasi-quantum groups and modularization Definition: Modular tensor category Definition Let C be a finite abelian k -linear tensor category. If C has rigid structure b V : V ∨ ⊗ V → I , d V : I → V ⊗ V ∨
Motivation Modular tensor categories Small quasi-quantum groups and modularization Definition: Modular tensor category Definition Let C be a finite abelian k -linear tensor category. If C has rigid structure b V : V ∨ ⊗ V → I , d V : I → V ⊗ V ∨ braiding c V , W : V ⊗ W → W ⊗ V
Motivation Modular tensor categories Small quasi-quantum groups and modularization Definition: Modular tensor category Definition Let C be a finite abelian k -linear tensor category. If C has rigid structure b V : V ∨ ⊗ V → I , d V : I → V ⊗ V ∨ braiding c V , W : V ⊗ W → W ⊗ V ribbon structure θ V : V → V ,
Motivation Modular tensor categories Small quasi-quantum groups and modularization Definition: Modular tensor category Definition Let C be a finite abelian k -linear tensor category. If C has rigid structure b V : V ∨ ⊗ V → I , d V : I → V ⊗ V ∨ braiding c V , W : V ⊗ W → W ⊗ V ribbon structure θ V : V → V , then C is called premodular .
Motivation Modular tensor categories Small quasi-quantum groups and modularization Definition: Modular tensor category Definition Let C be a finite abelian k -linear tensor category. If C has rigid structure b V : V ∨ ⊗ V → I , d V : I → V ⊗ V ∨ braiding c V , W : V ⊗ W → W ⊗ V ribbon structure θ V : V → V , then C is called premodular . If the braiding is non-degenerate, i.e. V ∼ = I n , c V , W ◦ c W , V = id V ⊗ W ∀ W ⇔ then C is called modular .
Montag, 3. September 2018 08:26 Montag, 3. September 2018 08:26 Motivation Modular tensor categories Small quasi-quantum groups and modularization Semisimple MTCs: SL (2 , Z )-action Semisimple modular tensor categories (MTCs) carry projective SL (2 , Z )-action: ( S -matrix) S �− → ( T -matrix) T �− → ( δ ij · θ i ) i , j ∈ I
Montag, 3. September 2018 08:26 Montag, 3. September 2018 08:26 Motivation Modular tensor categories Small quasi-quantum groups and modularization Semisimple MTCs: SL (2 , Z )-action Semisimple modular tensor categories (MTCs) carry projective SL (2 , Z )-action: ( S -matrix) S �− → ( T -matrix) T �− → ( δ ij · θ i ) i , j ∈ I More generally, semisimple MTCs yield 1 invariants of oriented, closed 3-manifolds, 2 projective representations of the mapping class groups of closed oriented surfaces, 3 3d TFTs from MTCs [Reshetikin,Turaev’91][Turaev’94]
Montag, 3. September 2018 08:26 Montag, 3. September 2018 08:26 Motivation Modular tensor categories Small quasi-quantum groups and modularization Semisimple MTCs: SL (2 , Z )-action Semisimple modular tensor categories (MTCs) carry projective SL (2 , Z )-action: ( S -matrix) S �− → ( T -matrix) T �− → ( δ ij · θ i ) i , j ∈ I More generally, semisimple MTCs yield 1 invariants of oriented, closed 3-manifolds, 2 projective representations of the mapping class groups of closed oriented surfaces, 3 3d TFTs from MTCs [Reshetikin,Turaev’91][Turaev’94] [ Lyubashenko ′ 95]: still true if we drop semisimplicity!
Motivation Modular tensor categories Small quasi-quantum groups and modularization Modularization: semisimple case What if a premodular category is not modular?
Motivation Modular tensor categories Small quasi-quantum groups and modularization Modularization: semisimple case What if a premodular category is not modular? Semisimple case: Definition (Bruguieres’00) Let C premodular, D modular. A dominant ribbon functor F : C → D is called a modularization of C .
Motivation Modular tensor categories Small quasi-quantum groups and modularization Modularization: semisimple case What if a premodular category is not modular? Semisimple case: Definition (Bruguieres’00) Let C premodular, D modular. A dominant ribbon functor F : C → D is called a modularization of C . Theorem (Bruguieres’00,Mueger’00) Let C premodular with trivial twist on transparent objects. Then a modularization of C exists. Proof relies strongly on Deligne’s theorem! Modularization is unique, have explicit construction!
Motivation Modular tensor categories Small quasi-quantum groups and modularization Example: � G -graded vector spaces Let C = kG - mod for G finite abelian group.
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