Non-semisimple modular tensor categories from …...as modular tensor categories for some nite...

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Non-semisimple modular tensor categories fromquasi-quantum groups

Tobias Ohrmann

Leibniz University Hannover

August 06, 2019


Motivation

Modular tensor categories (MTCs) are central objects in

1 2d conformal field theory (2d CFT): chiral half is encodedby representation category of underlying vertex operatoralgebra (VOA)

2 low-dimensional topology: Modular tensor categories yield

invariants of oriented closed 3-manifoldsmore generally: 3d topological field theories[Reshitikin,Turaev][Turaev,Viro]

3 FFRS-construction: chiral CFT + special symmetricFrobenius algebra in corresponding MTC ⇒ full CFT


Motivation







Motivation







Motivation







Motivation

Above results are

1 proven only in the rational case:

Def: VOA V rational :⇔ V-Rep is finite + semisimpleHuang’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

Above results are


Def: VOA V rational :⇔ V-Rep is finite + semisimple

Huang’04: V-Rep is rational MTCZhu’96: characters of V are modular invariant




Motivation

Above results are







Motivation

Above results are







Motivation

Above results are







Motivation

Above results are







Example: W(p)-algebras

[Kausch’91][Flohr’96][PRZ’06][FGST’06],...:Family of logarithmic CFTs associated to Virasoro (p, 1)-minimalmodels: W(p)-algebras

General construction:

Input data: finite dim. simple complex simply lacedLie algebra g, 2pth root of unit q

[Feigin,Tipunin’10]: General approach to constructnon-semisimple vertex algebra Wg(p) from this data

[Feigin,Tipunin’10][Adamovic,Milas’14],...:

Conjecture: Wg(p)-mod ∼= u-mod (1)

as modular tensor categories for some finite dim. factorizableribbon quasi-Hopf algebra u.




















Definition: Modular tensor category

Definition

Let C be a finite abelian k-linear tensor category. If C has

rigid structure bV : V ∨ ⊗ V → I, dV : I→ V ⊗ V ∨

braiding cV ,W : V ⊗W →W ⊗ V

ribbon structure θV : V → V ,

then C is called premodular. If the braiding is non-degenerate, i.e.

cV ,W ◦ cW ,V = idV⊗W ∀W ⇔ V ∼= In,

then C is called modular.



Definition










Definition










Definition










Definition





then C is called premodular.

If the braiding is non-degenerate, i.e.





Definition









Semisimple MTCs: SL(2,Z)-action

Semisimple modular tensor categories (MTCs) carryprojective SL(2,Z)-action:

S 7−→

Montag, 3. September 2018 08:26

(S-matrix)

T 7−→ (δij · θi )i ,j∈I


(T -matrix)

More generally, semisimple MTCs yield

1 invariants of oriented, closed 3-manifolds,

2 projective representations of the mapping class groups ofclosed oriented surfaces,

3 3d TFTs from MTCs [Reshetikin,Turaev’91][Turaev’94]

[Lyubashenko ′95]: still true if we drop semisimplicity!




S 7−→


(S-matrix)

T 7−→ (δij · θi )i ,j∈I


(T -matrix)









S 7−→


(S-matrix)

T 7−→ (δij · θi )i ,j∈I


(T -matrix)







Modularization: semisimple case

What if a premodular category is not modular?

Semisimple case:

Definition (Bruguieres’00)

Let C premodular, D modular. A dominant ribbon functorF : C → D is called a modularization of C.

Theorem (Bruguieres’00,Mueger’00)

Let C premodular with trivial twist on transparent objects. Then amodularization of C exists.

Proof relies strongly on Deligne’s theorem!Modularization is unique, have explicit construction!




Semisimple case:









Semisimple case:







Example: G -graded vector spaces

Let C = kG -mod for G finite abelian group.

kG -mod can be identified with G -graded vector spaces VectG

.

Definition (MacLane’50)

An abelian 3-cocycle (ω, σ) ∈ Z 3ab(G , k×) on an abelian group G is

a 3-cocycle ω ∈ Z 3(G , k×) together with 2-cochain σ ∈ C 2(G , k×)satisfying the hexagon equations.

MacLane: Abelian cohomology theory, H3ab(G , k×) ∼= QF (G , k×)

Proposition (Joyal-Street’86,DGNO’10)

Up to ribbon equivalence, ribbon structures on VectG areparametrised by H3

ab(G , k×)⊕ Hom(G ,±1).





.







ab(G , k×)⊕ Hom(G ,±1).





.







ab(G , k×)⊕ Hom(G ,±1).





.







ab(G , k×)⊕ Hom(G ,±1).



For (ω, σ) ∈ Z 3ab(G , k×), define symmetric bihomomorphism

B(χ, ψ) := σ(χ, ψ)σ(ψ, χ).

Lemma (Gainutdinov,Lentner,O.)

Vect(ω,σ,η)

Gmodular iff T := Rad(B) = 0.

A modularization of Vect(ω,σ,η)

Gexists if and only if

Q(τ) := σ(τ, τ) = 1, η(τ) = 1 ∀τ ∈ T .

Then, we call (ω, σ, η) modularizable ⇒ VectG/T

modular.

Explicit (ω, σ) ∈ Z 3ab(G/T ) for every section s : G/T → G



For (ω, σ) ∈ Z 3ab(G , k×), define symmetric bihomomorphism

B(χ, ψ) := σ(χ, ψ)σ(ψ, χ).

Lemma (Gainutdinov,Lentner,O.)

Vect(ω,σ,η)

Gmodular iff T := Rad(B) = 0.

A modularization of Vect(ω,σ,η)

Gexists if and only if

Q(τ) := σ(τ, τ) = 1, η(τ) = 1 ∀τ ∈ T .

Then, we call (ω, σ, η) modularizable ⇒ VectG/T

modular.

Explicit (ω, σ) ∈ Z 3ab(G/T ) for every section s : G/T → G


Modularization: non-semisimple case

Definition (Lyubashenko’95)

A premodular category C is called modular if the Hopf pairingωC : KC ⊗KC → I on the coend KC is non-degenerate.

Theorem (Shimizu’16)

A premodular category C is modular iff transparent objects aretrivial.

We propose the definition of a non-semisimple modularization:

Definition (Gainutdinov,Lentner,O.’18)

Let C premodular, D modular. A ribbon functor F : C → D iscalled a modularization of C if F (KC/Rad(ωC)) ∼= KD as braidedHopf algebras.




















(Quasi-)Hopf algebras

Finite dimensional ribbon quasi-Hopf algebras

Morally: Finite dimensional k-algebra H with additional structure,s.t. RepkH is a premodular category.

In particular,

coproduct ∆ : H → H ⊗ H induces tensor structure

coassociator φ ∈ H ⊗ H ⊗ H induces associator

R-matrix R ∈ H ⊗ H induces braiding

Ribbon element ν ∈ H induces ribbon structure

...

If RepkH is even modular, we call H factorizable.


(Quasi-)Hopf algebras

Finite dimensional ribbon quasi-Hopf algebras

Morally: Finite dimensional k-algebra H with additional structure,s.t. RepkH is a premodular category. In particular,

coproduct ∆ : H → H ⊗ H induces tensor structure

coassociator φ ∈ H ⊗ H ⊗ H induces associator

R-matrix R ∈ H ⊗ H induces braiding

Ribbon element ν ∈ H induces ribbon structure

...

If RepkH is even modular, we call H factorizable.


Example: Small quantum groups

[Drinfeld’85][Jimbo’85]: Fix f.d. semisimple complex Lie algebra g⇒ U(g) can be deformed to Hopf algebra Uq(g) (quantum group)

[Lusztig’90]: Set q to a root of unity

1 ⇒ Uq(g) has non-semisimple representation theory

2 Surjective map π : Uq(g) � U(g)⇒ ker(π) generated by augmentation ideal of fin. dim.sub-Hopf algebra uq(g) (small quantum group)

3 Ansatz for R-matrix: R = R0Θ ∈ uq(g)⊗ uq(g)

Here: Extend uq(g) by enlarging underlying group algebraC[ΛR/Λ′R ] via intermediate lattice ΛR ⊆ Λ ⊆ ΛW ⇒ uq(g,Λ)


















Example: W-algebras pt.2

We continue with the vertex algebra Wg(p) and discuss previousconjecture.

Case g = sl2: Equivalence as C-linear categories proven forsmall quantum group u = uq(sl2) [Nagatomo,Tsuchyia’09]

BUT: uq(sl2) does not allow for R-matrix

⇒ No braiding for u-mod!

[Creutzig,Gainutdinov,Runkel’17]: Modify coproduct onuq(sl2) via coassociator φ ⇒ quasi-Hopf algebra uφ

⇒ u(φ)-mod is non-semisimple modular tensor category

Still not clear if Conjecture holds





























Small quasi-quantum groups and modularization

[Lentner,O’17]:

list of all possible solutions of Lusztig ansatz for uq(g,Λ)

group of transparent objects

ribbon structure

[Gainutdinov,Lentner,O’18]:

Input data: finite abelian group G , abelian 3-cocycle(ω, σ) ∈ Z 3

ab(G ), χi ∈ G , 1 ≤ i ≤ n

Define V := ⊕ni=1 Vi ∈ Vect

(ω,σ)

G, assume that Nichols alg.

B(V ) ∈ Hopf(

Vect(ω,σ)

G

)is finite dimensional

Theorem (Gainutdinov,Lentner,O.’18)

Given the above data, the vector space B(V )⊗ kG ⊗ B(V ∗) canbe endowed with the structure of a quasi-triangular quasiHopf-algebra u(ω, σ) satifying the following relations:



[Lentner,O’17]:



ribbon structure



ab(G ), χi ∈ G , 1 ≤ i ≤ n


(ω,σ)


B(V ) ∈ Hopf(

Vect(ω,σ)

G






[Lentner,O’17]:



ribbon structure



ab(G ), χi ∈ G , 1 ≤ i ≤ n


(ω,σ)


B(V ) ∈ Hopf(

Vect(ω,σ)

G






[Lentner,O’17]:



ribbon structure



ab(G ), χi ∈ G , 1 ≤ i ≤ n


(ω,σ)


B(V ) ∈ Hopf(

Vect(ω,σ)

G





Relations of u(ω, σ)

Generators: Kχ, Kψ for χ, ψ ∈ G , Ei ,Fj for 1 ≤ i , j ≤ n.

Kχψ = θχ,ψKχKψ Kχψθχ,ψ = KχKψ

KχEiθχχi

= σ(χ, χi )EiKχ KχEi = σ(χi , χ)Ei Kχθχχi

KχFiθχχi = σ(χ, χi )FiKχ KχFiθ

χχi = σ(χ, χi )FiKχθ

χχi

EiFj − QijFjEi = δij(Ki − K−1j )

KχKχ is grouplike

If ω = 1 ⇒ Red elements vanish and Kχ = Kχ.


Quantum Serre relations

Although ω 6= 1, we can associate to V = ⊕ni=1 Vi ∈ Vect

(ω,σ)

Ga

diagonally braided vector space (|V |, qij := σ(χi , χj)).

Lemma

The quantum Serre relations for B(V ) and B(|V |) are the same.

For a general braided abelian monoidal category C, we have thefollowing:

Theorem (O.)

The Woronowicz symmetrizer of the adjoint actionWc ◦ adn : V⊗n ⊗W → V⊗n ⊗W is given by

Wc ◦ adn =n−1∏j=0

(id⊗(n+1)

−(id⊗(n−1) ⊗ c2

)◦ ((cn−1,n ◦ · · · ◦ cn−j ,n−j+1)⊗ id)

).


Quantum Serre relations

Although ω 6= 1, we can associate to V = ⊕ni=1 Vi ∈ Vect

(ω,σ)

Ga

diagonally braided vector space (|V |, qij := σ(χi , χj)).

Lemma

The quantum Serre relations for B(V ) and B(|V |) are the same.

For a general braided abelian monoidal category C, we have thefollowing:

Theorem (O.)

The Woronowicz symmetrizer of the adjoint actionWc ◦ adn : V⊗n ⊗W → V⊗n ⊗W is given by

Wc ◦ adn =n−1∏j=0

(id⊗(n+1)

−(id⊗(n−1) ⊗ c2

)◦ ((cn−1,n ◦ · · · ◦ cn−j ,n−j+1)⊗ id)

).


Modularization of u-mod

Recall: grouplike elements of the small quantum group u are givenby G = Λ/Λ′.


Let u be an ordinary small quantum group with R-matrixR = R0Θ, s.t. the corresponding tuple (ω = 1, σ) on G ismodularizable. Then,

∃ subgroup G ⊆ G , datum (ω, σ, χi ∈ G ), twist J ∈ u ⊗ u,s.t.

u := u(ω, σ) ↪→ uJ

is a quasi-Hopf inclusion.

the restriction functor F : u-mod → u-mod is amodularization.



Proof: Need to show that u-mod is modular iff kG -mod (ω,σ) ismodular. In [GLO’18]: Defined Verma module functor

V : Vect(ω,σ)

G→ u(ω, σ)-mod for small quasi-quantum groups

⇒ V braided colax monoidal.

More elegantly:

[Shimizu’16] BBYD(C)′ ∼= C′

For ordinary small quantum groups u = B(V )⊗ kG ⊗ B(V ∗):B(V )B(V )YD(kG -mod) ∼= u-mod

Work in progress: This is still true for a large class of quasi-Hopfalgebras, such as u(ω, σ).



Proof: Need to show that u-mod is modular iff kG -mod (ω,σ) ismodular. In [GLO’18]: Defined Verma module functor

V : Vect(ω,σ)

G→ u(ω, σ)-mod for small quasi-quantum groups

⇒ V braided colax monoidal.More elegantly:

[Shimizu’16] BBYD(C)′ ∼= C′

For ordinary small quantum groups u = B(V )⊗ kG ⊗ B(V ∗):B(V )B(V )YD(kG -mod) ∼= u-mod

Work in progress: This is still true for a large class of quasi-Hopfalgebras, such as u(ω, σ).


Thank you for your attention!

Date post:	08-Jul-2020
Category:	Documents
Upload:	others
View:	2 times
Download:	0 times

Non-semisimple modular tensor categories from …...as modular tensor categories for some nite...

Documents