non-semisimple modular tensor categories from …...as modular tensor categories for some nite...
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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 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 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 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 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 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 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 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 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 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 + 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 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 MTCZhu’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
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 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 + 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 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 + 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 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 + 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 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)-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.
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)-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.
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)-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.
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
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
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
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
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
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) carryprojective SL(2,Z)-action:
S 7−→
Montag, 3. September 2018 08:26
(S-matrix)
T 7−→ (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 ofclosed 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
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
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 ofclosed 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
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
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 ofclosed 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?
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!
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 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!
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 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!
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 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).
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 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).
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 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).
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 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).
Motivation Modular tensor categories Small quasi-quantum groups and modularization
Example: G -graded vector spaces
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
Motivation Modular tensor categories Small quasi-quantum groups and modularization
Example: G -graded vector spaces
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
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 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.
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 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.
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 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.
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
(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)
[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)
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 = 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)
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 = 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)
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 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
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 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
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 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
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 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
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 := ⊕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:
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 := ⊕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:
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 := ⊕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:
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 := ⊕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:
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 = 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.
Motivation Modular tensor categories Small quasi-quantum groups and modularization
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)
).
Motivation Modular tensor categories Small quasi-quantum groups and modularization
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)
).
Motivation Modular tensor categories Small quasi-quantum groups and modularization
Modularization of u-mod
Recall: grouplike elements of the small quantum group u are givenby G = Λ/Λ′.
Theorem (Gainutdinov,Lentner,O.’18)
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.
Motivation Modular tensor categories Small quasi-quantum groups and modularization
Modularization of u-mod
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).
Motivation Modular tensor categories Small quasi-quantum groups and modularization
Modularization of u-mod
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).
Motivation Modular tensor categories Small quasi-quantum groups and modularization
Thank you for your attention!