Non-semisimple modular tensor categories from quasi-quantum groups - - PowerPoint PPT Presentation

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, 2pth root of unit q [Feigin,Tipunin’10]: General approach to construct non-semisimple vertex algebra Wg(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, 2pth root of unit q [Feigin,Tipunin’10]: General approach to construct non-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. 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 bV : V ∨ ⊗ V → I, dV : 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 bV : V ∨ ⊗ V → I, dV : I → V ⊗ V ∨ braiding cV ,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 bV : V ∨ ⊗ V → I, dV : I → V ⊗ V ∨ braiding cV ,W : V ⊗ W → W ⊗ V ribbon structure θV : 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 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.

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 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.

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 − →

Montag, 3. September 2018 08:26

(S-matrix) T − → (δij · θi)i,j∈I

Montag, 3. September 2018 08:26

(T-matrix)

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 − →

Montag, 3. September 2018 08:26

(S-matrix) T − → (δij · θi)i,j∈I

Montag, 3. September 2018 08:26

(T-matrix) 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]

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 − →

Montag, 3. September 2018 08:26

(S-matrix) T − → (δij · θi)i,j∈I

Montag, 3. September 2018 08:26

(T-matrix) 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.

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Example: G-graded vector spaces

Let C = kG-mod for G finite abelian group. kG-mod can be identified with G-graded vector spaces Vect

G.

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Example: G-graded vector spaces

Let C = kG-mod for G finite abelian group. kG-mod can be identified with G-graded vector spaces Vect

G.

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Example: G-graded vector spaces

Let C = kG-mod for G finite abelian group. kG-mod can be identified with G-graded vector spaces Vect

G.

Definition (MacLane’50) An abelian 3-cocycle (ω, σ) ∈ Z 3

ab(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, H3

ab(G, k×) ∼

= QF(G, k×) Proposition (Joyal-Street’86,DGNO’10) Up to ribbon equivalence, ribbon structures on VectG are parametrised by H3

ab(G, k×) ⊕ Hom(G, ±1).

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Example: G-graded vector spaces

For (ω, σ) ∈ Z 3

ab(

G, k×), define symmetric bihomomorphism B(χ, ψ) := σ(χ, ψ)σ(ψ, χ).

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Example: G-graded vector spaces

For (ω, σ) ∈ Z 3

ab(

G, k×), define symmetric bihomomorphism B(χ, ψ) := σ(χ, ψ)σ(ψ, χ). Lemma (Gainutdinov,Lentner,O.) Vect(ω,σ,η)

• G

modular iff T := Rad(B) = 0. A modularization of Vect(ω,σ,η)

• G

exists if and only if Q(τ) := σ(τ, τ) = 1, η(τ) = 1 ∀τ ∈ T. Then, we call (ω, σ, η) modularizable ⇒ Vect

G/T modular.

Explicit (¯ ω, ¯ σ) ∈ Z 3

ab(

G/T) for every section s : G/T → G

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Modularization: non-semisimple case

Definition (Lyubashenko’95) A premodular category C is called modular if the Hopf pairing ωC : KC ⊗ KC → I on the coend KC is non-degenerate.

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Modularization: non-semisimple case

Definition (Lyubashenko’95) A premodular category C is called modular if the Hopf pairing ωC : KC ⊗ KC → I on the coend KC is non-degenerate. Theorem (Shimizu’16) A premodular category C is modular iff transparent objects are trivial.

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Modularization: non-semisimple case

Definition (Lyubashenko’95) A premodular category C is called modular if the Hopf pairing ωC : KC ⊗ KC → I on the coend KC is non-degenerate. Theorem (Shimizu’16) A premodular category C is modular iff transparent objects are trivial. 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 is called a modularization of C if F(KC/Rad(ωC)) ∼ = KD as braided Hopf algebras.

Motivation Modular tensor categories Small quasi-quantum groups and modularization

(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.

Motivation Modular tensor categories Small quasi-quantum groups and modularization

(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.

Motivation Modular tensor categories Small quasi-quantum groups and modularization

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)

Motivation Modular tensor categories Small quasi-quantum groups and modularization

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 = R0 ¯

Θ ∈ uq(g) ⊗ uq(g)

Motivation Modular tensor categories Small quasi-quantum groups and modularization

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 = R0 ¯

Θ ∈ uq(g) ⊗ uq(g) Here: Extend uq(g) by enlarging underlying group algebra C[ΛR/Λ′

R] via intermediate lattice ΛR ⊆ Λ ⊆ ΛW ⇒ uq(g, Λ)

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Example: W-algebras pt.2

We continue with the vertex algebra Wg(p) and discuss previous conjecture. Case g = sl2: Equivalence as C-linear categories proven for small quantum group u = uq(sl2) [Nagatomo,Tsuchyia’09]

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Example: W-algebras pt.2

We continue with the vertex algebra Wg(p) and discuss previous conjecture. Case g = sl2: Equivalence as C-linear categories proven for small quantum group u = uq(sl2) [Nagatomo,Tsuchyia’09] BUT: uq(sl2) does not allow for R-matrix ⇒ No braiding for u-mod!

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Example: W-algebras pt.2

We continue with the vertex algebra Wg(p) and discuss previous conjecture. Case g = sl2: Equivalence as C-linear categories proven for small 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 on uq(sl2) via coassociator φ ⇒ quasi-Hopf algebra uφ ⇒ u(φ)-mod is non-semisimple modular tensor category

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Example: W-algebras pt.2

We continue with the vertex algebra Wg(p) and discuss previous conjecture. Case g = sl2: Equivalence as C-linear categories proven for small 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 on uq(sl2) via coassociator φ ⇒ quasi-Hopf algebra uφ ⇒ u(φ)-mod is non-semisimple modular tensor category Still not clear if Conjecture holds

Motivation Modular tensor categories Small quasi-quantum groups and modularization

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

Motivation Modular tensor categories Small quasi-quantum groups and modularization

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

Motivation Modular tensor categories Small quasi-quantum groups and modularization

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 := ⊕n

i=1 Vi ∈ Vect(ω,σ)

• G

, assume that Nichols alg. B(V ) ∈ Hopf

• Vect(ω,σ)
• G
• is finite dimensional

Motivation Modular tensor categories Small quasi-quantum groups and modularization

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 := ⊕n

i=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 ∗) can be endowed with the structure of a quasi-triangular quasi Hopf-algebra u(ω, σ) satifying the following relations:

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Relations of u(ω, σ)

Generators: Kχ, ¯ Kψ for χ, ψ ∈ G, Ei, Fj for 1 ≤ i, j ≤ n. Kχψ = θχ,ψKχKψ ¯ Kχψθχ,ψ = ¯ Kχ ¯ Kψ KχEiθχ

¯ χi = σ(¯

χ, χi)EiKχ ¯ KχEi = σ(χi, ¯ χ)Ei ¯ Kχθχ

¯ χi

KχFiθχ

χi = σ(¯

χ, ¯ χi)FiKχ ¯ KχFiθχ

χi = σ(¯

χ, ¯ χi)FiKχθχ

χi

EiFj − QijFjEi = δij(Ki − ¯ K −1

j

) ¯ KχKχ is grouplike If ω = 1 ⇒ Red elements vanish and ¯ Kχ = Kχ.

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Quantum Serre relations

Although ω = 1, we can associate to V = ⊕n

i=1 Vi ∈ Vect(ω,σ)

• G

a diagonally braided vector space (|V |, qij := σ(¯ χi, ¯ χj)). Lemma The quantum Serre relations for B(V ) and B(|V |) are the same.

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Quantum Serre relations

Although ω = 1, we can associate to V = ⊕n

i=1 Vi ∈ Vect(ω,σ)

• G

a 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 the following: Theorem (O.) The Woronowicz symmetrizer of the adjoint action Wc ◦ 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)
• .

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Modularization of u-mod

Recall: grouplike elements of the small quantum group u are given by G = Λ/Λ′. Theorem (Gainutdinov,Lentner,O.’18) Let u be an ordinary small quantum group with R-matrix R = R0 ¯ Θ, s.t. the corresponding tuple (ω = 1, σ) on G is

• modularizable. 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 a modularization.

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Modularization of u-mod

Proof: Need to show that ¯ u-mod is modular iff k ¯ G-mod(¯

ω,¯ σ) is

• modular. 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.

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Modularization of u-mod

Proof: Need to show that ¯ u-mod is modular iff k ¯ G-mod(¯

ω,¯ σ) is

• modular. 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] B

BYD(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-Hopf algebras, such as u(ω, σ).

Motivation Modular tensor categories Small quasi-quantum groups and modularization

Thank you for your attention!