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Motivations Fusion category and braided fusion category Main theorems Applications
Classification of fusion categories andfactorizable semisimple Hopf algebras
Zhiqiang YuJoint with Victor Ostrik
Yangzhou University
2021. 8. 23
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Motivations Fusion category and braided fusion category Main theorems Applications
1 Motivations
2 Fusion category and braided fusion category
3 Main theorems
4 Applications
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Motivations Fusion category and braided fusion category Main theorems Applications
Motivations
M. Muger defined Morita equivalence of fusion categories[J. Pure Appl. Algebra, 2003], and proposed the minimalmodular extension conjecture [Proc. Lond. Math. Soc,2003].T. Lan, L. Kong, X. Wen studied the minimal extensionconjecture from the aspect of topological orders [Comm.Phys. Math, 2017].In [Adv. Math, 2011], P. Etingof, D. Nikshych, V. Ostrikdefined weakly group-theoretical fusion categories andsolvable fusion categories, which are closely related finitegroups, (quasi-)Hopf algebras.
![Page 4: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/4.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Motivations
M. Muger defined Morita equivalence of fusion categories[J. Pure Appl. Algebra, 2003], and proposed the minimalmodular extension conjecture [Proc. Lond. Math. Soc,2003].T. Lan, L. Kong, X. Wen studied the minimal extensionconjecture from the aspect of topological orders [Comm.Phys. Math, 2017].In [Adv. Math, 2011], P. Etingof, D. Nikshych, V. Ostrikdefined weakly group-theoretical fusion categories andsolvable fusion categories, which are closely related finitegroups, (quasi-)Hopf algebras.
![Page 5: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/5.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Motivations
M. Muger defined Morita equivalence of fusion categories[J. Pure Appl. Algebra, 2003], and proposed the minimalmodular extension conjecture [Proc. Lond. Math. Soc,2003].T. Lan, L. Kong, X. Wen studied the minimal extensionconjecture from the aspect of topological orders [Comm.Phys. Math, 2017].In [Adv. Math, 2011], P. Etingof, D. Nikshych, V. Ostrikdefined weakly group-theoretical fusion categories andsolvable fusion categories, which are closely related finitegroups, (quasi-)Hopf algebras.
![Page 6: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/6.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Let K be a field, K = K, char(K) = 0.
DefinitionC is a finite tensor category if:
C ∼= Rep(A) as a K-linear abelian category, where A is af-d unital associative K-algebra, Rep(A) is the category off-d left A-modules.C = (C, I,⊗, ρ, λ, α) is a monoidal category, and −⊗− is aK-linear exact bifunctor.Every object X has a left dual X ∗ and right dual ∗X.I is a simple object.
DefinitionC is a fusion category, if it is a semisimple finite tensor category.
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Motivations Fusion category and braided fusion category Main theorems Applications
Let K be a field, K = K, char(K) = 0.
DefinitionC is a finite tensor category if:
C ∼= Rep(A) as a K-linear abelian category, where A is af-d unital associative K-algebra, Rep(A) is the category off-d left A-modules.C = (C, I,⊗, ρ, λ, α) is a monoidal category, and −⊗− is aK-linear exact bifunctor.Every object X has a left dual X ∗ and right dual ∗X.I is a simple object.
DefinitionC is a fusion category, if it is a semisimple finite tensor category.
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Motivations Fusion category and braided fusion category Main theorems Applications
Let K be a field, K = K, char(K) = 0.
DefinitionC is a finite tensor category if:
C ∼= Rep(A) as a K-linear abelian category, where A is af-d unital associative K-algebra, Rep(A) is the category off-d left A-modules.C = (C, I,⊗, ρ, λ, α) is a monoidal category, and −⊗− is aK-linear exact bifunctor.Every object X has a left dual X ∗ and right dual ∗X.I is a simple object.
DefinitionC is a fusion category, if it is a semisimple finite tensor category.
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Motivations Fusion category and braided fusion category Main theorems Applications
Let K be a field, K = K, char(K) = 0.
DefinitionC is a finite tensor category if:
C ∼= Rep(A) as a K-linear abelian category, where A is af-d unital associative K-algebra, Rep(A) is the category off-d left A-modules.C = (C, I,⊗, ρ, λ, α) is a monoidal category, and −⊗− is aK-linear exact bifunctor.Every object X has a left dual X ∗ and right dual ∗X.I is a simple object.
DefinitionC is a fusion category, if it is a semisimple finite tensor category.
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Motivations Fusion category and braided fusion category Main theorems Applications
Let K be a field, K = K, char(K) = 0.
DefinitionC is a finite tensor category if:
C ∼= Rep(A) as a K-linear abelian category, where A is af-d unital associative K-algebra, Rep(A) is the category off-d left A-modules.C = (C, I,⊗, ρ, λ, α) is a monoidal category, and −⊗− is aK-linear exact bifunctor.Every object X has a left dual X ∗ and right dual ∗X.I is a simple object.
DefinitionC is a fusion category, if it is a semisimple finite tensor category.
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Motivations Fusion category and braided fusion category Main theorems Applications
Let K be a field, K = K, char(K) = 0.
DefinitionC is a finite tensor category if:
C ∼= Rep(A) as a K-linear abelian category, where A is af-d unital associative K-algebra, Rep(A) is the category off-d left A-modules.C = (C, I,⊗, ρ, λ, α) is a monoidal category, and −⊗− is aK-linear exact bifunctor.Every object X has a left dual X ∗ and right dual ∗X.I is a simple object.
DefinitionC is a fusion category, if it is a semisimple finite tensor category.
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Motivations Fusion category and braided fusion category Main theorems Applications
Let K be a field, K = K, char(K) = 0.
DefinitionC is a finite tensor category if:
C ∼= Rep(A) as a K-linear abelian category, where A is af-d unital associative K-algebra, Rep(A) is the category off-d left A-modules.C = (C, I,⊗, ρ, λ, α) is a monoidal category, and −⊗− is aK-linear exact bifunctor.Every object X has a left dual X ∗ and right dual ∗X.I is a simple object.
DefinitionC is a fusion category, if it is a semisimple finite tensor category.
![Page 13: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/13.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Let K be a field, K = K, char(K) = 0.
DefinitionC is a finite tensor category if:
C ∼= Rep(A) as a K-linear abelian category, where A is af-d unital associative K-algebra, Rep(A) is the category off-d left A-modules.C = (C, I,⊗, ρ, λ, α) is a monoidal category, and −⊗− is aK-linear exact bifunctor.Every object X has a left dual X ∗ and right dual ∗X.I is a simple object.
DefinitionC is a fusion category, if it is a semisimple finite tensor category.
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Motivations Fusion category and braided fusion category Main theorems Applications
Examples of fusion categories
Vec, the category of finite-dimensional vector spaces, withtensor product ⊗K over field K.Pointed fusion category VecωG, the category of G-gradedfinite-dimensional vector spaces category, where G is afinite group, ω ∈ Z 3(G,K∗) is a normalized 3-cocycle,K∗ := K\{0}. Explicitly, αVg1 ,Vg2 ,Vg3
= ω(g1,g2,g3) :
(Vg1 ⊗ Vg2)⊗ Vg3
∼→ Vg1 ⊗ (Vg2 ⊗ Vg3).Let G be a finite group, then Rep(G) is a fusion category.Let H be a f-d (quasi-)Hopf algebra, then Rep(H) is a finitetensor category. Rep(H) is a fusion category iff H issemisimple (quasi-)Hopf algebra.
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Motivations Fusion category and braided fusion category Main theorems Applications
Examples of fusion categories
Vec, the category of finite-dimensional vector spaces, withtensor product ⊗K over field K.Pointed fusion category VecωG, the category of G-gradedfinite-dimensional vector spaces category, where G is afinite group, ω ∈ Z 3(G,K∗) is a normalized 3-cocycle,K∗ := K\{0}. Explicitly, αVg1 ,Vg2 ,Vg3
= ω(g1,g2,g3) :
(Vg1 ⊗ Vg2)⊗ Vg3
∼→ Vg1 ⊗ (Vg2 ⊗ Vg3).Let G be a finite group, then Rep(G) is a fusion category.Let H be a f-d (quasi-)Hopf algebra, then Rep(H) is a finitetensor category. Rep(H) is a fusion category iff H issemisimple (quasi-)Hopf algebra.
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Motivations Fusion category and braided fusion category Main theorems Applications
Examples of fusion categories
Vec, the category of finite-dimensional vector spaces, withtensor product ⊗K over field K.Pointed fusion category VecωG, the category of G-gradedfinite-dimensional vector spaces category, where G is afinite group, ω ∈ Z 3(G,K∗) is a normalized 3-cocycle,K∗ := K\{0}. Explicitly, αVg1 ,Vg2 ,Vg3
= ω(g1,g2,g3) :
(Vg1 ⊗ Vg2)⊗ Vg3
∼→ Vg1 ⊗ (Vg2 ⊗ Vg3).Let G be a finite group, then Rep(G) is a fusion category.Let H be a f-d (quasi-)Hopf algebra, then Rep(H) is a finitetensor category. Rep(H) is a fusion category iff H issemisimple (quasi-)Hopf algebra.
![Page 17: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/17.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Examples of fusion categories
Vec, the category of finite-dimensional vector spaces, withtensor product ⊗K over field K.Pointed fusion category VecωG, the category of G-gradedfinite-dimensional vector spaces category, where G is afinite group, ω ∈ Z 3(G,K∗) is a normalized 3-cocycle,K∗ := K\{0}. Explicitly, αVg1 ,Vg2 ,Vg3
= ω(g1,g2,g3) :
(Vg1 ⊗ Vg2)⊗ Vg3
∼→ Vg1 ⊗ (Vg2 ⊗ Vg3).Let G be a finite group, then Rep(G) is a fusion category.Let H be a f-d (quasi-)Hopf algebra, then Rep(H) is a finitetensor category. Rep(H) is a fusion category iff H issemisimple (quasi-)Hopf algebra.
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Motivations Fusion category and braided fusion category Main theorems Applications
Frobenius-Perron dimension
Definition (ENO, Ann. of. Math, 2005)
Let Gr(C) be the Grothendieck ring of fusion category C. Thenthere exists a unique homomorphism FPdim(−) : Gr(C)→ Ksuch that FPdim(X ) is a positive algebraic integer, ∀X ∈ O(C).
Definition
The sum FPdim(C) :=∑
X∈O(C) FPdim(X )2 is defined as theFrobenius-Perron dimension of C.
Example
C = Rep(H), where H is a s.s (quasi-)Hopf algebra. ThenFPdim(X ) = dimK(X ) for X ∈ O(C). In particular,
FPdim(C) =∑
X∈O(C)
FPdim(X )2 = dimK(H)
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Motivations Fusion category and braided fusion category Main theorems Applications
Frobenius-Perron dimension
Definition (ENO, Ann. of. Math, 2005)
Let Gr(C) be the Grothendieck ring of fusion category C. Thenthere exists a unique homomorphism FPdim(−) : Gr(C)→ Ksuch that FPdim(X ) is a positive algebraic integer, ∀X ∈ O(C).
Definition
The sum FPdim(C) :=∑
X∈O(C) FPdim(X )2 is defined as theFrobenius-Perron dimension of C.
Example
C = Rep(H), where H is a s.s (quasi-)Hopf algebra. ThenFPdim(X ) = dimK(X ) for X ∈ O(C). In particular,
FPdim(C) =∑
X∈O(C)
FPdim(X )2 = dimK(H)
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Motivations Fusion category and braided fusion category Main theorems Applications
Frobenius-Perron dimension
Definition (ENO, Ann. of. Math, 2005)
Let Gr(C) be the Grothendieck ring of fusion category C. Thenthere exists a unique homomorphism FPdim(−) : Gr(C)→ Ksuch that FPdim(X ) is a positive algebraic integer, ∀X ∈ O(C).
Definition
The sum FPdim(C) :=∑
X∈O(C) FPdim(X )2 is defined as theFrobenius-Perron dimension of C.
Example
C = Rep(H), where H is a s.s (quasi-)Hopf algebra. ThenFPdim(X ) = dimK(X ) for X ∈ O(C). In particular,
FPdim(C) =∑
X∈O(C)
FPdim(X )2 = dimK(H)
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Motivations Fusion category and braided fusion category Main theorems Applications
Frobenius-Perron dimension
Definition (ENO, Ann. of. Math, 2005)
Let Gr(C) be the Grothendieck ring of fusion category C. Thenthere exists a unique homomorphism FPdim(−) : Gr(C)→ Ksuch that FPdim(X ) is a positive algebraic integer, ∀X ∈ O(C).
Definition
The sum FPdim(C) :=∑
X∈O(C) FPdim(X )2 is defined as theFrobenius-Perron dimension of C.
Example
C = Rep(H), where H is a s.s (quasi-)Hopf algebra. ThenFPdim(X ) = dimK(X ) for X ∈ O(C). In particular,
FPdim(C) =∑
X∈O(C)
FPdim(X )2 = dimK(H)
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Motivations Fusion category and braided fusion category Main theorems Applications
Integral fusion categories
DefinitionA fusion category C is integral if FPdim(X ) ∈ Z, ∀X ∈ O(C).
Theorem (ENO, Ann. of. Math, 2005)
A fusion category C is integral iff C ∼= Rep(H) for some s.squasi-Hopf algebra H.
DefinitionA fusion category C is weakly integral if FPdim(C) ∈ Z.
ExampleLet d ∈ Z≥3 be square-free. Then Tambara-Yamagami fusioncategory C := T Y(Zd , χ, ν) is weakly integral: O(C) = {X}∪Zd ,
X ⊗ X = ⊕g∈Zd g, g ⊗ X = X ⊗ g = X .
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Motivations Fusion category and braided fusion category Main theorems Applications
Integral fusion categories
DefinitionA fusion category C is integral if FPdim(X ) ∈ Z, ∀X ∈ O(C).
Theorem (ENO, Ann. of. Math, 2005)
A fusion category C is integral iff C ∼= Rep(H) for some s.squasi-Hopf algebra H.
DefinitionA fusion category C is weakly integral if FPdim(C) ∈ Z.
ExampleLet d ∈ Z≥3 be square-free. Then Tambara-Yamagami fusioncategory C := T Y(Zd , χ, ν) is weakly integral: O(C) = {X}∪Zd ,
X ⊗ X = ⊕g∈Zd g, g ⊗ X = X ⊗ g = X .
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Motivations Fusion category and braided fusion category Main theorems Applications
Integral fusion categories
DefinitionA fusion category C is integral if FPdim(X ) ∈ Z, ∀X ∈ O(C).
Theorem (ENO, Ann. of. Math, 2005)
A fusion category C is integral iff C ∼= Rep(H) for some s.squasi-Hopf algebra H.
DefinitionA fusion category C is weakly integral if FPdim(C) ∈ Z.
ExampleLet d ∈ Z≥3 be square-free. Then Tambara-Yamagami fusioncategory C := T Y(Zd , χ, ν) is weakly integral: O(C) = {X}∪Zd ,
X ⊗ X = ⊕g∈Zd g, g ⊗ X = X ⊗ g = X .
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Motivations Fusion category and braided fusion category Main theorems Applications
Integral fusion categories
DefinitionA fusion category C is integral if FPdim(X ) ∈ Z, ∀X ∈ O(C).
Theorem (ENO, Ann. of. Math, 2005)
A fusion category C is integral iff C ∼= Rep(H) for some s.squasi-Hopf algebra H.
DefinitionA fusion category C is weakly integral if FPdim(C) ∈ Z.
ExampleLet d ∈ Z≥3 be square-free. Then Tambara-Yamagami fusioncategory C := T Y(Zd , χ, ν) is weakly integral: O(C) = {X}∪Zd ,
X ⊗ X = ⊕g∈Zd g, g ⊗ X = X ⊗ g = X .
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Motivations Fusion category and braided fusion category Main theorems Applications
Nilpotent fusion categories
Definition (ENO, Ann. of. Math, 2005)Let C be a fusion category. The adjoint subcategory Cad of C isthe fusion category generated by object X ∈ O(C) such that
X ⊆ Y ⊗ Y ∗ for some Y ∈ O(C).
Definition (S. Gelaki, D. Nikshych, Adv. Math, 2008)
Fusion category C is nilpotent, if C(n) = Vec, where C(0) := C,C(1) := Cad, C(m) := C(m−1)
ad , m ∈ Z≥1.
Example
Let G be a finite group, Z (G) the center of G, C = Rep(G).Then Cad = Rep(G/Z (G)). Thus, C is nilpotent iff G is nilpotent.
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Motivations Fusion category and braided fusion category Main theorems Applications
Nilpotent fusion categories
Definition (ENO, Ann. of. Math, 2005)Let C be a fusion category. The adjoint subcategory Cad of C isthe fusion category generated by object X ∈ O(C) such that
X ⊆ Y ⊗ Y ∗ for some Y ∈ O(C).
Definition (S. Gelaki, D. Nikshych, Adv. Math, 2008)
Fusion category C is nilpotent, if C(n) = Vec, where C(0) := C,C(1) := Cad, C(m) := C(m−1)
ad , m ∈ Z≥1.
Example
Let G be a finite group, Z (G) the center of G, C = Rep(G).Then Cad = Rep(G/Z (G)). Thus, C is nilpotent iff G is nilpotent.
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Motivations Fusion category and braided fusion category Main theorems Applications
Nilpotent fusion categories
Definition (ENO, Ann. of. Math, 2005)Let C be a fusion category. The adjoint subcategory Cad of C isthe fusion category generated by object X ∈ O(C) such that
X ⊆ Y ⊗ Y ∗ for some Y ∈ O(C).
Definition (S. Gelaki, D. Nikshych, Adv. Math, 2008)
Fusion category C is nilpotent, if C(n) = Vec, where C(0) := C,C(1) := Cad, C(m) := C(m−1)
ad , m ∈ Z≥1.
Example
Let G be a finite group, Z (G) the center of G, C = Rep(G).Then Cad = Rep(G/Z (G)). Thus, C is nilpotent iff G is nilpotent.
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Motivations Fusion category and braided fusion category Main theorems Applications
Nilpotent fusion categories
Definition (ENO, Ann. of. Math, 2005)Let C be a fusion category. The adjoint subcategory Cad of C isthe fusion category generated by object X ∈ O(C) such that
X ⊆ Y ⊗ Y ∗ for some Y ∈ O(C).
Definition (S. Gelaki, D. Nikshych, Adv. Math, 2008)
Fusion category C is nilpotent, if C(n) = Vec, where C(0) := C,C(1) := Cad, C(m) := C(m−1)
ad , m ∈ Z≥1.
Example
Let G be a finite group, Z (G) the center of G, C = Rep(G).Then Cad = Rep(G/Z (G)). Thus, C is nilpotent iff G is nilpotent.
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Motivations Fusion category and braided fusion category Main theorems Applications
Nilpotent fusion categories
Definition (ENO, Ann. of. Math, 2005)Let C be a fusion category. The adjoint subcategory Cad of C isthe fusion category generated by object X ∈ O(C) such that
X ⊆ Y ⊗ Y ∗ for some Y ∈ O(C).
Definition (S. Gelaki, D. Nikshych, Adv. Math, 2008)
Fusion category C is nilpotent, if C(n) = Vec, where C(0) := C,C(1) := Cad, C(m) := C(m−1)
ad , m ∈ Z≥1.
Example
Let G be a finite group, Z (G) the center of G, C = Rep(G).Then Cad = Rep(G/Z (G)). Thus, C is nilpotent iff G is nilpotent.
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Motivations Fusion category and braided fusion category Main theorems Applications
Weakly group-theoretical fusion categories
Remark (S. Gelaki, D. Nikshych, Adv. Math, 2008)Indeed, C is nilpotent iff there exist a series of fusionsubcategories C0 = Vec ⊆ C1 ⊆ · · · ⊆ Cn = C such that Cj is aGj -graded extension of Cj−1, where Gj are finite groups.
Theorem (ENO, Adv. Math, 2011)Let C be a fusion category.
C is group-theoretical iff Z(C) ∼= Z(VecωG).C is weakly group-theoretical iff Z(C) ∼= Z(D), where D is anilpotent fusion category.
Question (ENO. Adv. Math, 2011)Are weakly integral fusion categories weakly group-theoretical?
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Motivations Fusion category and braided fusion category Main theorems Applications
Weakly group-theoretical fusion categories
Remark (S. Gelaki, D. Nikshych, Adv. Math, 2008)Indeed, C is nilpotent iff there exist a series of fusionsubcategories C0 = Vec ⊆ C1 ⊆ · · · ⊆ Cn = C such that Cj is aGj -graded extension of Cj−1, where Gj are finite groups.
Theorem (ENO, Adv. Math, 2011)Let C be a fusion category.
C is group-theoretical iff Z(C) ∼= Z(VecωG).C is weakly group-theoretical iff Z(C) ∼= Z(D), where D is anilpotent fusion category.
Question (ENO. Adv. Math, 2011)Are weakly integral fusion categories weakly group-theoretical?
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Motivations Fusion category and braided fusion category Main theorems Applications
Weakly group-theoretical fusion categories
Remark (S. Gelaki, D. Nikshych, Adv. Math, 2008)Indeed, C is nilpotent iff there exist a series of fusionsubcategories C0 = Vec ⊆ C1 ⊆ · · · ⊆ Cn = C such that Cj is aGj -graded extension of Cj−1, where Gj are finite groups.
Theorem (ENO, Adv. Math, 2011)Let C be a fusion category.
C is group-theoretical iff Z(C) ∼= Z(VecωG).C is weakly group-theoretical iff Z(C) ∼= Z(D), where D is anilpotent fusion category.
Question (ENO. Adv. Math, 2011)Are weakly integral fusion categories weakly group-theoretical?
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Motivations Fusion category and braided fusion category Main theorems Applications
Weakly group-theoretical fusion categories
Remark (S. Gelaki, D. Nikshych, Adv. Math, 2008)Indeed, C is nilpotent iff there exist a series of fusionsubcategories C0 = Vec ⊆ C1 ⊆ · · · ⊆ Cn = C such that Cj is aGj -graded extension of Cj−1, where Gj are finite groups.
Theorem (ENO, Adv. Math, 2011)Let C be a fusion category.
C is group-theoretical iff Z(C) ∼= Z(VecωG).C is weakly group-theoretical iff Z(C) ∼= Z(D), where D is anilpotent fusion category.
Question (ENO. Adv. Math, 2011)Are weakly integral fusion categories weakly group-theoretical?
![Page 35: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/35.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Weakly group-theoretical fusion categories
Remark (S. Gelaki, D. Nikshych, Adv. Math, 2008)Indeed, C is nilpotent iff there exist a series of fusionsubcategories C0 = Vec ⊆ C1 ⊆ · · · ⊆ Cn = C such that Cj is aGj -graded extension of Cj−1, where Gj are finite groups.
Theorem (ENO, Adv. Math, 2011)Let C be a fusion category.
C is group-theoretical iff Z(C) ∼= Z(VecωG).C is weakly group-theoretical iff Z(C) ∼= Z(D), where D is anilpotent fusion category.
Question (ENO. Adv. Math, 2011)Are weakly integral fusion categories weakly group-theoretical?
![Page 36: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/36.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Weakly group-theoretical fusion categories
Remark (S. Gelaki, D. Nikshych, Adv. Math, 2008)Indeed, C is nilpotent iff there exist a series of fusionsubcategories C0 = Vec ⊆ C1 ⊆ · · · ⊆ Cn = C such that Cj is aGj -graded extension of Cj−1, where Gj are finite groups.
Theorem (ENO, Adv. Math, 2011)Let C be a fusion category.
C is group-theoretical iff Z(C) ∼= Z(VecωG).C is weakly group-theoretical iff Z(C) ∼= Z(D), where D is anilpotent fusion category.
Question (ENO. Adv. Math, 2011)Are weakly integral fusion categories weakly group-theoretical?
![Page 37: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/37.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Frobenius property
Theorem (ENO. Adv. Math, 2011)A weakly group-theoretical fusion category C admits Frobeniusproperty.That is, FPdim(C)
FPdim(X) is an algebraic integer, ∀X ∈ O(C).
RemarkIf the previous conjecture is true, then the Kaplansky’ssixth conjecture is true.Weakly group-theoretical property is invariant under takingDrinfeld centers, (de-)equivariantizations, G-extensions.
QuestionAre the Drinfeld centers of weakly integral fusion categoriesweakly group-theoretical?
![Page 38: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/38.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Frobenius property
Theorem (ENO. Adv. Math, 2011)A weakly group-theoretical fusion category C admits Frobeniusproperty.That is, FPdim(C)
FPdim(X) is an algebraic integer, ∀X ∈ O(C).
RemarkIf the previous conjecture is true, then the Kaplansky’ssixth conjecture is true.Weakly group-theoretical property is invariant under takingDrinfeld centers, (de-)equivariantizations, G-extensions.
QuestionAre the Drinfeld centers of weakly integral fusion categoriesweakly group-theoretical?
![Page 39: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/39.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Frobenius property
Theorem (ENO. Adv. Math, 2011)A weakly group-theoretical fusion category C admits Frobeniusproperty.That is, FPdim(C)
FPdim(X) is an algebraic integer, ∀X ∈ O(C).
RemarkIf the previous conjecture is true, then the Kaplansky’ssixth conjecture is true.Weakly group-theoretical property is invariant under takingDrinfeld centers, (de-)equivariantizations, G-extensions.
QuestionAre the Drinfeld centers of weakly integral fusion categoriesweakly group-theoretical?
![Page 40: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/40.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Frobenius property
Theorem (ENO. Adv. Math, 2011)A weakly group-theoretical fusion category C admits Frobeniusproperty.That is, FPdim(C)
FPdim(X) is an algebraic integer, ∀X ∈ O(C).
RemarkIf the previous conjecture is true, then the Kaplansky’ssixth conjecture is true.Weakly group-theoretical property is invariant under takingDrinfeld centers, (de-)equivariantizations, G-extensions.
QuestionAre the Drinfeld centers of weakly integral fusion categoriesweakly group-theoretical?
![Page 41: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/41.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Frobenius property
Theorem (ENO. Adv. Math, 2011)A weakly group-theoretical fusion category C admits Frobeniusproperty.That is, FPdim(C)
FPdim(X) is an algebraic integer, ∀X ∈ O(C).
RemarkIf the previous conjecture is true, then the Kaplansky’ssixth conjecture is true.Weakly group-theoretical property is invariant under takingDrinfeld centers, (de-)equivariantizations, G-extensions.
QuestionAre the Drinfeld centers of weakly integral fusion categoriesweakly group-theoretical?
![Page 42: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/42.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Braided fusion category
DefinitionFusion category C is a braided fusion category if there exists anatural isomorphism cX ,Y : X ⊗ Y ∼→ Y ⊗ X, X ,Y ∈ C satisfyingthe braiding equations.
Example
C = Rep(G), where the braiding c of Rep(G) is given bythe reflection τ of vector spaces. We call C a Tannakiancategory.C = Rep(G, z), where z ∈ Z (G) and z2 = 1, the braiding ofRep(G, z) is given by τ ◦ R with
R =12
(1⊗ 1 + z ⊗ 1 + 1⊗ z − z ⊗ z).
![Page 43: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/43.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Braided fusion category
DefinitionFusion category C is a braided fusion category if there exists anatural isomorphism cX ,Y : X ⊗ Y ∼→ Y ⊗ X, X ,Y ∈ C satisfyingthe braiding equations.
Example
C = Rep(G), where the braiding c of Rep(G) is given bythe reflection τ of vector spaces. We call C a Tannakiancategory.C = Rep(G, z), where z ∈ Z (G) and z2 = 1, the braiding ofRep(G, z) is given by τ ◦ R with
R =12
(1⊗ 1 + z ⊗ 1 + 1⊗ z − z ⊗ z).
![Page 44: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/44.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Braided fusion category
DefinitionFusion category C is a braided fusion category if there exists anatural isomorphism cX ,Y : X ⊗ Y ∼→ Y ⊗ X, X ,Y ∈ C satisfyingthe braiding equations.
Example
C = Rep(G), where the braiding c of Rep(G) is given bythe reflection τ of vector spaces. We call C a Tannakiancategory.C = Rep(G, z), where z ∈ Z (G) and z2 = 1, the braiding ofRep(G, z) is given by τ ◦ R with
R =12
(1⊗ 1 + z ⊗ 1 + 1⊗ z − z ⊗ z).
![Page 45: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/45.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Braided fusion category
DefinitionFusion category C is a braided fusion category if there exists anatural isomorphism cX ,Y : X ⊗ Y ∼→ Y ⊗ X, X ,Y ∈ C satisfyingthe braiding equations.
Example
C = Rep(G), where the braiding c of Rep(G) is given bythe reflection τ of vector spaces. We call C a Tannakiancategory.C = Rep(G, z), where z ∈ Z (G) and z2 = 1, the braiding ofRep(G, z) is given by τ ◦ R with
R =12
(1⊗ 1 + z ⊗ 1 + 1⊗ z − z ⊗ z).
![Page 46: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/46.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Muger centralizer
Definition (M. Muger, Adv. Math, 2000)Let C ⊆ D be braided fusion categories.The following braidedfusion category
D′C := {Y ∈ C|cY ,X cX ,Y = idX⊗Y ,∀X ∈ D}
is called the Muger centralizer of C in D. Denote the Mugercenter of C by C′ := C′C .
DefinitionLet C be a braided fusion category.
C is non-degenerate if C′ = Vec.C is slightly degenerate if C′ = sVec := Rep(Z2, z).C is symmetric if C′ = C.
![Page 47: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/47.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Muger centralizer
Definition (M. Muger, Adv. Math, 2000)Let C ⊆ D be braided fusion categories.The following braidedfusion category
D′C := {Y ∈ C|cY ,X cX ,Y = idX⊗Y ,∀X ∈ D}
is called the Muger centralizer of C in D. Denote the Mugercenter of C by C′ := C′C .
DefinitionLet C be a braided fusion category.
C is non-degenerate if C′ = Vec.C is slightly degenerate if C′ = sVec := Rep(Z2, z).C is symmetric if C′ = C.
![Page 48: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/48.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Muger centralizer
Definition (M. Muger, Adv. Math, 2000)Let C ⊆ D be braided fusion categories.The following braidedfusion category
D′C := {Y ∈ C|cY ,X cX ,Y = idX⊗Y ,∀X ∈ D}
is called the Muger centralizer of C in D. Denote the Mugercenter of C by C′ := C′C .
DefinitionLet C be a braided fusion category.
C is non-degenerate if C′ = Vec.C is slightly degenerate if C′ = sVec := Rep(Z2, z).C is symmetric if C′ = C.
![Page 49: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/49.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Muger centralizer
Definition (M. Muger, Adv. Math, 2000)Let C ⊆ D be braided fusion categories.The following braidedfusion category
D′C := {Y ∈ C|cY ,X cX ,Y = idX⊗Y ,∀X ∈ D}
is called the Muger centralizer of C in D. Denote the Mugercenter of C by C′ := C′C .
DefinitionLet C be a braided fusion category.
C is non-degenerate if C′ = Vec.C is slightly degenerate if C′ = sVec := Rep(Z2, z).C is symmetric if C′ = C.
![Page 50: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/50.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Muger centralizer
Definition (M. Muger, Adv. Math, 2000)Let C ⊆ D be braided fusion categories.The following braidedfusion category
D′C := {Y ∈ C|cY ,X cX ,Y = idX⊗Y ,∀X ∈ D}
is called the Muger centralizer of C in D. Denote the Mugercenter of C by C′ := C′C .
DefinitionLet C be a braided fusion category.
C is non-degenerate if C′ = Vec.C is slightly degenerate if C′ = sVec := Rep(Z2, z).C is symmetric if C′ = C.
![Page 51: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/51.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Muger centralizer
Definition (M. Muger, Adv. Math, 2000)Let C ⊆ D be braided fusion categories.The following braidedfusion category
D′C := {Y ∈ C|cY ,X cX ,Y = idX⊗Y ,∀X ∈ D}
is called the Muger centralizer of C in D. Denote the Mugercenter of C by C′ := C′C .
DefinitionLet C be a braided fusion category.
C is non-degenerate if C′ = Vec.C is slightly degenerate if C′ = sVec := Rep(Z2, z).C is symmetric if C′ = C.
![Page 52: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/52.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Muger centralizer
Definition (M. Muger, Adv. Math, 2000)Let C ⊆ D be braided fusion categories.The following braidedfusion category
D′C := {Y ∈ C|cY ,X cX ,Y = idX⊗Y ,∀X ∈ D}
is called the Muger centralizer of C in D. Denote the Mugercenter of C by C′ := C′C .
DefinitionLet C be a braided fusion category.
C is non-degenerate if C′ = Vec.C is slightly degenerate if C′ = sVec := Rep(Z2, z).C is symmetric if C′ = C.
![Page 53: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/53.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Muger centralizer
Definition (M. Muger, Adv. Math, 2000)Let C ⊆ D be braided fusion categories.The following braidedfusion category
D′C := {Y ∈ C|cY ,X cX ,Y = idX⊗Y ,∀X ∈ D}
is called the Muger centralizer of C in D. Denote the Mugercenter of C by C′ := C′C .
DefinitionLet C be a braided fusion category.
C is non-degenerate if C′ = Vec.C is slightly degenerate if C′ = sVec := Rep(Z2, z).C is symmetric if C′ = C.
![Page 54: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/54.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Example
Let C be a fusion category, then the Drinfeld center Z(C) isnon-degenerate.Let k = 4m + 2 ∈ Z≥2 and C := C(sl2, k + 2,q) withq2 = e
2πik+2 , then C is non-degenerate, while Cad is slightly
degenerate.
Theorem (P. Deligne, Moscow Math. J, 2002)Any symmetric fusion category C is braided equivalent toRep(G, z). In particular, C contains a maximal Tannakiansubcategory Rep(G/〈z〉).
![Page 55: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/55.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Example
Let C be a fusion category, then the Drinfeld center Z(C) isnon-degenerate.Let k = 4m + 2 ∈ Z≥2 and C := C(sl2, k + 2,q) withq2 = e
2πik+2 , then C is non-degenerate, while Cad is slightly
degenerate.
Theorem (P. Deligne, Moscow Math. J, 2002)Any symmetric fusion category C is braided equivalent toRep(G, z). In particular, C contains a maximal Tannakiansubcategory Rep(G/〈z〉).
![Page 56: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/56.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Example
Let C be a fusion category, then the Drinfeld center Z(C) isnon-degenerate.Let k = 4m + 2 ∈ Z≥2 and C := C(sl2, k + 2,q) withq2 = e
2πik+2 , then C is non-degenerate, while Cad is slightly
degenerate.
Theorem (P. Deligne, Moscow Math. J, 2002)Any symmetric fusion category C is braided equivalent toRep(G, z). In particular, C contains a maximal Tannakiansubcategory Rep(G/〈z〉).
![Page 57: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/57.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
DefinitionLet G be a finite abelian group. η : G→ K∗ is a quadratic formon G, if
η(g) = η(g−1), ∀g ∈ G,
B(g,h) := η(gh)η(g)η(h) defines a symmetric bi-character.
η is non-degenerate if B(−,−) is.
Theorem (A. Joyal, R. Street, Adv. Math, 1993)There exists a bijective correspondence between metric groupsand pointed non-degenerate fusion categories.
We denote the category determined by metric group (G, η) byC(G, η) below.
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Motivations Fusion category and braided fusion category Main theorems Applications
DefinitionLet G be a finite abelian group. η : G→ K∗ is a quadratic formon G, if
η(g) = η(g−1), ∀g ∈ G,
B(g,h) := η(gh)η(g)η(h) defines a symmetric bi-character.
η is non-degenerate if B(−,−) is.
Theorem (A. Joyal, R. Street, Adv. Math, 1993)There exists a bijective correspondence between metric groupsand pointed non-degenerate fusion categories.
We denote the category determined by metric group (G, η) byC(G, η) below.
![Page 59: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/59.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
DefinitionLet G be a finite abelian group. η : G→ K∗ is a quadratic formon G, if
η(g) = η(g−1), ∀g ∈ G,
B(g,h) := η(gh)η(g)η(h) defines a symmetric bi-character.
η is non-degenerate if B(−,−) is.
Theorem (A. Joyal, R. Street, Adv. Math, 1993)There exists a bijective correspondence between metric groupsand pointed non-degenerate fusion categories.
We denote the category determined by metric group (G, η) byC(G, η) below.
![Page 60: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/60.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
DefinitionLet G be a finite abelian group. η : G→ K∗ is a quadratic formon G, if
η(g) = η(g−1), ∀g ∈ G,
B(g,h) := η(gh)η(g)η(h) defines a symmetric bi-character.
η is non-degenerate if B(−,−) is.
Theorem (A. Joyal, R. Street, Adv. Math, 1993)There exists a bijective correspondence between metric groupsand pointed non-degenerate fusion categories.
We denote the category determined by metric group (G, η) byC(G, η) below.
![Page 61: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/61.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
DefinitionLet G be a finite abelian group. η : G→ K∗ is a quadratic formon G, if
η(g) = η(g−1), ∀g ∈ G,
B(g,h) := η(gh)η(g)η(h) defines a symmetric bi-character.
η is non-degenerate if B(−,−) is.
Theorem (A. Joyal, R. Street, Adv. Math, 1993)There exists a bijective correspondence between metric groupsand pointed non-degenerate fusion categories.
We denote the category determined by metric group (G, η) byC(G, η) below.
![Page 62: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/62.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
DefinitionLet G be a finite abelian group. η : G→ K∗ is a quadratic formon G, if
η(g) = η(g−1), ∀g ∈ G,
B(g,h) := η(gh)η(g)η(h) defines a symmetric bi-character.
η is non-degenerate if B(−,−) is.
Theorem (A. Joyal, R. Street, Adv. Math, 1993)There exists a bijective correspondence between metric groupsand pointed non-degenerate fusion categories.
We denote the category determined by metric group (G, η) byC(G, η) below.
![Page 63: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/63.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Main theorem
Theorem (V. Ostrik, Z. Yu, arXiv:2105.0181)
Let n,d ∈ Z≥1, (n,d) = 1 and d is square-free. Assume that Cis a non-degenerate fusion category with FPdim(C) = nd. If
C contains a Tannakian category E := Rep(G) such that(E ′C)G
∼= A� C(Zd , η) as braided fusion category,(FPdim(X )2,d) = 1 for all X ∈ O(C),
then C ∼= C(Zd , η) � C(Zd , η)′C as braided fusion category.
Remark(E ′C)G is the de-equivariantization of E ′C by E .The condition (FPdim(X )2,d) = 1 is necessary. AssumeC := T Y(Zd , χ, µ)Z2 , then C contains two simple objects ofFP-dimension
√d, but C does not contain C(Zd , η).
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Motivations Fusion category and braided fusion category Main theorems Applications
Main theorem
Theorem (V. Ostrik, Z. Yu, arXiv:2105.0181)
Let n,d ∈ Z≥1, (n,d) = 1 and d is square-free. Assume that Cis a non-degenerate fusion category with FPdim(C) = nd. If
C contains a Tannakian category E := Rep(G) such that(E ′C)G
∼= A� C(Zd , η) as braided fusion category,(FPdim(X )2,d) = 1 for all X ∈ O(C),
then C ∼= C(Zd , η) � C(Zd , η)′C as braided fusion category.
Remark(E ′C)G is the de-equivariantization of E ′C by E .The condition (FPdim(X )2,d) = 1 is necessary. AssumeC := T Y(Zd , χ, µ)Z2 , then C contains two simple objects ofFP-dimension
√d, but C does not contain C(Zd , η).
![Page 65: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/65.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Main theorem
Theorem (V. Ostrik, Z. Yu, arXiv:2105.0181)
Let n,d ∈ Z≥1, (n,d) = 1 and d is square-free. Assume that Cis a non-degenerate fusion category with FPdim(C) = nd. If
C contains a Tannakian category E := Rep(G) such that(E ′C)G
∼= A� C(Zd , η) as braided fusion category,(FPdim(X )2,d) = 1 for all X ∈ O(C),
then C ∼= C(Zd , η) � C(Zd , η)′C as braided fusion category.
Remark(E ′C)G is the de-equivariantization of E ′C by E .The condition (FPdim(X )2,d) = 1 is necessary. AssumeC := T Y(Zd , χ, µ)Z2 , then C contains two simple objects ofFP-dimension
√d, but C does not contain C(Zd , η).
![Page 66: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/66.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Main theorem
Theorem (V. Ostrik, Z. Yu, arXiv:2105.0181)
Let n,d ∈ Z≥1, (n,d) = 1 and d is square-free. Assume that Cis a non-degenerate fusion category with FPdim(C) = nd. If
C contains a Tannakian category E := Rep(G) such that(E ′C)G
∼= A� C(Zd , η) as braided fusion category,(FPdim(X )2,d) = 1 for all X ∈ O(C),
then C ∼= C(Zd , η) � C(Zd , η)′C as braided fusion category.
Remark(E ′C)G is the de-equivariantization of E ′C by E .The condition (FPdim(X )2,d) = 1 is necessary. AssumeC := T Y(Zd , χ, µ)Z2 , then C contains two simple objects ofFP-dimension
√d, but C does not contain C(Zd , η).
![Page 67: Classification of fusion categories and factorizable](https://reader031.vdocuments.net/reader031/viewer/2022021804/620d7aece0249623a269156c/html5/thumbnails/67.jpg)
Motivations Fusion category and braided fusion category Main theorems Applications
Main theorem
Theorem (V. Ostrik, Z. Yu, arXiv:2105.0181)
Let n,d ∈ Z≥1, (n,d) = 1 and d is square-free. Assume that Cis a non-degenerate fusion category with FPdim(C) = nd. If
C contains a Tannakian category E := Rep(G) such that(E ′C)G
∼= A� C(Zd , η) as braided fusion category,(FPdim(X )2,d) = 1 for all X ∈ O(C),
then C ∼= C(Zd , η) � C(Zd , η)′C as braided fusion category.
Remark(E ′C)G is the de-equivariantization of E ′C by E .The condition (FPdim(X )2,d) = 1 is necessary. AssumeC := T Y(Zd , χ, µ)Z2 , then C contains two simple objects ofFP-dimension
√d, but C does not contain C(Zd , η).
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Motivations Fusion category and braided fusion category Main theorems Applications
Main theorem
Theorem (V. Ostrik, Z. Yu, arXiv:2105.0181)
Let n,d ∈ Z≥1, (n,d) = 1 and d is square-free. Assume that Cis a non-degenerate fusion category with FPdim(C) = nd. If
C contains a Tannakian category E := Rep(G) such that(E ′C)G
∼= A� C(Zd , η) as braided fusion category,(FPdim(X )2,d) = 1 for all X ∈ O(C),
then C ∼= C(Zd , η) � C(Zd , η)′C as braided fusion category.
Remark(E ′C)G is the de-equivariantization of E ′C by E .The condition (FPdim(X )2,d) = 1 is necessary. AssumeC := T Y(Zd , χ, µ)Z2 , then C contains two simple objects ofFP-dimension
√d, but C does not contain C(Zd , η).
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Motivations Fusion category and braided fusion category Main theorems Applications
Main theorem
Theorem (V. Ostrik, Z. Yu, arXiv:2105.0181)
Let n,d ∈ Z≥1, (n,d) = 1 and d is square-free. Assume that Cis a non-degenerate fusion category with FPdim(C) = nd. If
C contains a Tannakian category E := Rep(G) such that(E ′C)G
∼= A� C(Zd , η) as braided fusion category,(FPdim(X )2,d) = 1 for all X ∈ O(C),
then C ∼= C(Zd , η) � C(Zd , η)′C as braided fusion category.
Remark(E ′C)G is the de-equivariantization of E ′C by E .The condition (FPdim(X )2,d) = 1 is necessary. AssumeC := T Y(Zd , χ, µ)Z2 , then C contains two simple objects ofFP-dimension
√d, but C does not contain C(Zd , η).
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Motivations Fusion category and braided fusion category Main theorems Applications
Corollary (V. Ostrik, Z. Yu, arXiv:2105.0181)Let C be an integral non-degenerate fusion category withFPdim(C) = nd, n,d ∈ Z≥1, (n,d) = 1 and d is square-free.
If C is weakly group-theoretical, there exists a braidedequivalence C ∼= C(Zd , η) � C(Zd , η)′C ;If n = paqb, where p,q are primes, then C is solvable. Inparticular, C ∼= C(Zd , η) � C(Zd , η)′C .If n = pa, then C is nilpotent and group-theoretical.
RemarkThe subcase that n = pa with p being an odd prime was provedin [J. Dong, S. Natale, Algebr. Represent. Theory, 2018].
Corollary (Z. Yu, J. Algebra, 2020)
Braided fusion categories of FP-dimensions paqbd are weaklygroup-theoretical, p,q are primes and (pq,d) = 1.
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Motivations Fusion category and braided fusion category Main theorems Applications
Corollary (V. Ostrik, Z. Yu, arXiv:2105.0181)Let C be an integral non-degenerate fusion category withFPdim(C) = nd, n,d ∈ Z≥1, (n,d) = 1 and d is square-free.
If C is weakly group-theoretical, there exists a braidedequivalence C ∼= C(Zd , η) � C(Zd , η)′C ;If n = paqb, where p,q are primes, then C is solvable. Inparticular, C ∼= C(Zd , η) � C(Zd , η)′C .If n = pa, then C is nilpotent and group-theoretical.
RemarkThe subcase that n = pa with p being an odd prime was provedin [J. Dong, S. Natale, Algebr. Represent. Theory, 2018].
Corollary (Z. Yu, J. Algebra, 2020)
Braided fusion categories of FP-dimensions paqbd are weaklygroup-theoretical, p,q are primes and (pq,d) = 1.
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Motivations Fusion category and braided fusion category Main theorems Applications
Corollary (V. Ostrik, Z. Yu, arXiv:2105.0181)Let C be an integral non-degenerate fusion category withFPdim(C) = nd, n,d ∈ Z≥1, (n,d) = 1 and d is square-free.
If C is weakly group-theoretical, there exists a braidedequivalence C ∼= C(Zd , η) � C(Zd , η)′C ;If n = paqb, where p,q are primes, then C is solvable. Inparticular, C ∼= C(Zd , η) � C(Zd , η)′C .If n = pa, then C is nilpotent and group-theoretical.
RemarkThe subcase that n = pa with p being an odd prime was provedin [J. Dong, S. Natale, Algebr. Represent. Theory, 2018].
Corollary (Z. Yu, J. Algebra, 2020)
Braided fusion categories of FP-dimensions paqbd are weaklygroup-theoretical, p,q are primes and (pq,d) = 1.
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Motivations Fusion category and braided fusion category Main theorems Applications
Corollary (V. Ostrik, Z. Yu, arXiv:2105.0181)Let C be an integral non-degenerate fusion category withFPdim(C) = nd, n,d ∈ Z≥1, (n,d) = 1 and d is square-free.
If C is weakly group-theoretical, there exists a braidedequivalence C ∼= C(Zd , η) � C(Zd , η)′C ;If n = paqb, where p,q are primes, then C is solvable. Inparticular, C ∼= C(Zd , η) � C(Zd , η)′C .If n = pa, then C is nilpotent and group-theoretical.
RemarkThe subcase that n = pa with p being an odd prime was provedin [J. Dong, S. Natale, Algebr. Represent. Theory, 2018].
Corollary (Z. Yu, J. Algebra, 2020)
Braided fusion categories of FP-dimensions paqbd are weaklygroup-theoretical, p,q are primes and (pq,d) = 1.
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Motivations Fusion category and braided fusion category Main theorems Applications
Corollary (V. Ostrik, Z. Yu, arXiv:2105.0181)Let C be an integral non-degenerate fusion category withFPdim(C) = nd, n,d ∈ Z≥1, (n,d) = 1 and d is square-free.
If C is weakly group-theoretical, there exists a braidedequivalence C ∼= C(Zd , η) � C(Zd , η)′C ;If n = paqb, where p,q are primes, then C is solvable. Inparticular, C ∼= C(Zd , η) � C(Zd , η)′C .If n = pa, then C is nilpotent and group-theoretical.
RemarkThe subcase that n = pa with p being an odd prime was provedin [J. Dong, S. Natale, Algebr. Represent. Theory, 2018].
Corollary (Z. Yu, J. Algebra, 2020)
Braided fusion categories of FP-dimensions paqbd are weaklygroup-theoretical, p,q are primes and (pq,d) = 1.
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Motivations Fusion category and braided fusion category Main theorems Applications
Corollary (V. Ostrik, Z. Yu, arXiv:2105.0181)Let C be an integral non-degenerate fusion category withFPdim(C) = nd, n,d ∈ Z≥1, (n,d) = 1 and d is square-free.
If C is weakly group-theoretical, there exists a braidedequivalence C ∼= C(Zd , η) � C(Zd , η)′C ;If n = paqb, where p,q are primes, then C is solvable. Inparticular, C ∼= C(Zd , η) � C(Zd , η)′C .If n = pa, then C is nilpotent and group-theoretical.
RemarkThe subcase that n = pa with p being an odd prime was provedin [J. Dong, S. Natale, Algebr. Represent. Theory, 2018].
Corollary (Z. Yu, J. Algebra, 2020)
Braided fusion categories of FP-dimensions paqbd are weaklygroup-theoretical, p,q are primes and (pq,d) = 1.
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Motivations Fusion category and braided fusion category Main theorems Applications
Minimal extension conjecture
Definition (M. Muger, Proc. Lond. Math. Soc, 2003)Let C be a braided fusion category. A non-degenerate fusioncategory D is a minimal extension of C if C ⊆ D and C′D ∼= C′.
RemarkIndeed, D is a minimal extension of C if and only if C ⊆ Dand FPdim(D) = FPdim(C)FPdim(C′).However, a braided fusion category may not admit aminimal extension (unpublished note by V. Drinfeld).
Theorem (V. Ostrik, Z. Yu, arXiv:2105.0181)Let C be a slightly degenerate fusion category. If C is Wittequivalent to a weakly group-theoretical fusion category, then Cadmits a minimal extension.
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Motivations Fusion category and braided fusion category Main theorems Applications
Minimal extension conjecture
Definition (M. Muger, Proc. Lond. Math. Soc, 2003)Let C be a braided fusion category. A non-degenerate fusioncategory D is a minimal extension of C if C ⊆ D and C′D ∼= C′.
RemarkIndeed, D is a minimal extension of C if and only if C ⊆ Dand FPdim(D) = FPdim(C)FPdim(C′).However, a braided fusion category may not admit aminimal extension (unpublished note by V. Drinfeld).
Theorem (V. Ostrik, Z. Yu, arXiv:2105.0181)Let C be a slightly degenerate fusion category. If C is Wittequivalent to a weakly group-theoretical fusion category, then Cadmits a minimal extension.
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Motivations Fusion category and braided fusion category Main theorems Applications
Minimal extension conjecture
Definition (M. Muger, Proc. Lond. Math. Soc, 2003)Let C be a braided fusion category. A non-degenerate fusioncategory D is a minimal extension of C if C ⊆ D and C′D ∼= C′.
RemarkIndeed, D is a minimal extension of C if and only if C ⊆ Dand FPdim(D) = FPdim(C)FPdim(C′).However, a braided fusion category may not admit aminimal extension (unpublished note by V. Drinfeld).
Theorem (V. Ostrik, Z. Yu, arXiv:2105.0181)Let C be a slightly degenerate fusion category. If C is Wittequivalent to a weakly group-theoretical fusion category, then Cadmits a minimal extension.
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Motivations Fusion category and braided fusion category Main theorems Applications
Minimal extension conjecture
Definition (M. Muger, Proc. Lond. Math. Soc, 2003)Let C be a braided fusion category. A non-degenerate fusioncategory D is a minimal extension of C if C ⊆ D and C′D ∼= C′.
RemarkIndeed, D is a minimal extension of C if and only if C ⊆ Dand FPdim(D) = FPdim(C)FPdim(C′).However, a braided fusion category may not admit aminimal extension (unpublished note by V. Drinfeld).
Theorem (V. Ostrik, Z. Yu, arXiv:2105.0181)Let C be a slightly degenerate fusion category. If C is Wittequivalent to a weakly group-theoretical fusion category, then Cadmits a minimal extension.
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Motivations Fusion category and braided fusion category Main theorems Applications
Corollary (V. Ostrik, Z. Yu, arXiv:2105.0181)
Assume C′ = sVec and FPdim(C) = nd, where n,d ∈ Z, 2 - dand (n,d) = 1. If C is weakly group-theoretical, and for anyobject X ∈ O(A), (FPdim(X )2,d) = 1, where A is a minimalextension of C. Then C ∼= C(Zd , η) � C(Zd , η)′C .
Corollary (V. Ostrik, Z. Yu, arXiv:2105.0181)
Assume C′ = sVec and FPdim(C) = nd, where n,d ∈ Z, 2 - dand (n,d) = 1. If C is integral and weakly group-theoretical,then C ∼= C(Zd , η) � C(Zd , η)′C .
Corollary (V. Ostrik, Z. Yu, arXiv:2015.0181; Z. Yu, J. Algebra,2020)
Assume C′ = sVec and FPdim(C) = pnd, where p is a prime,n,d ∈ Z and (n,d) = 1. If C is integral, then C is nilpotent andgroup-theoretical.
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Motivations Fusion category and braided fusion category Main theorems Applications
Corollary (V. Ostrik, Z. Yu, arXiv:2105.0181)
Assume C′ = sVec and FPdim(C) = nd, where n,d ∈ Z, 2 - dand (n,d) = 1. If C is weakly group-theoretical, and for anyobject X ∈ O(A), (FPdim(X )2,d) = 1, where A is a minimalextension of C. Then C ∼= C(Zd , η) � C(Zd , η)′C .
Corollary (V. Ostrik, Z. Yu, arXiv:2105.0181)
Assume C′ = sVec and FPdim(C) = nd, where n,d ∈ Z, 2 - dand (n,d) = 1. If C is integral and weakly group-theoretical,then C ∼= C(Zd , η) � C(Zd , η)′C .
Corollary (V. Ostrik, Z. Yu, arXiv:2015.0181; Z. Yu, J. Algebra,2020)
Assume C′ = sVec and FPdim(C) = pnd, where p is a prime,n,d ∈ Z and (n,d) = 1. If C is integral, then C is nilpotent andgroup-theoretical.
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Motivations Fusion category and braided fusion category Main theorems Applications
Corollary (V. Ostrik, Z. Yu, arXiv:2105.0181)
Assume C′ = sVec and FPdim(C) = nd, where n,d ∈ Z, 2 - dand (n,d) = 1. If C is weakly group-theoretical, and for anyobject X ∈ O(A), (FPdim(X )2,d) = 1, where A is a minimalextension of C. Then C ∼= C(Zd , η) � C(Zd , η)′C .
Corollary (V. Ostrik, Z. Yu, arXiv:2105.0181)
Assume C′ = sVec and FPdim(C) = nd, where n,d ∈ Z, 2 - dand (n,d) = 1. If C is integral and weakly group-theoretical,then C ∼= C(Zd , η) � C(Zd , η)′C .
Corollary (V. Ostrik, Z. Yu, arXiv:2015.0181; Z. Yu, J. Algebra,2020)
Assume C′ = sVec and FPdim(C) = pnd, where p is a prime,n,d ∈ Z and (n,d) = 1. If C is integral, then C is nilpotent andgroup-theoretical.
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Motivations Fusion category and braided fusion category Main theorems Applications
Factorizable Hopf algebra
DefinitionLet (H,R) be a f-d quasi-triangular Hopf algebra over K. H isfactorizable if the Drinfeld map Φ : H∗ → H is an isomorphism,where Φ(f ) = (id⊗ f )(R21R).
Example
Drinfeld double (D(H),R) is factorizable, where H is a f-dHopf algebra and R =
∑i(1⊗ hi)⊗ (hi ⊗ ε).
(K[Zd ],Rm) is a factorizable Hopf algebra with
Rm :=∑
0≤i,j≤n−1
ζ−ijdn
g i ⊗ g−mj ∈ K[Zd ]⊗K[Zd ],
where ζd is a primitive d-root of unity, 2 - m and (m,d) = 1.
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Motivations Fusion category and braided fusion category Main theorems Applications
Factorizable Hopf algebra
DefinitionLet (H,R) be a f-d quasi-triangular Hopf algebra over K. H isfactorizable if the Drinfeld map Φ : H∗ → H is an isomorphism,where Φ(f ) = (id⊗ f )(R21R).
Example
Drinfeld double (D(H),R) is factorizable, where H is a f-dHopf algebra and R =
∑i(1⊗ hi)⊗ (hi ⊗ ε).
(K[Zd ],Rm) is a factorizable Hopf algebra with
Rm :=∑
0≤i,j≤n−1
ζ−ijdn
g i ⊗ g−mj ∈ K[Zd ]⊗K[Zd ],
where ζd is a primitive d-root of unity, 2 - m and (m,d) = 1.
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Motivations Fusion category and braided fusion category Main theorems Applications
Factorizable Hopf algebra
DefinitionLet (H,R) be a f-d quasi-triangular Hopf algebra over K. H isfactorizable if the Drinfeld map Φ : H∗ → H is an isomorphism,where Φ(f ) = (id⊗ f )(R21R).
Example
Drinfeld double (D(H),R) is factorizable, where H is a f-dHopf algebra and R =
∑i(1⊗ hi)⊗ (hi ⊗ ε).
(K[Zd ],Rm) is a factorizable Hopf algebra with
Rm :=∑
0≤i,j≤n−1
ζ−ijdn
g i ⊗ g−mj ∈ K[Zd ]⊗K[Zd ],
where ζd is a primitive d-root of unity, 2 - m and (m,d) = 1.
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Motivations Fusion category and braided fusion category Main theorems Applications
Remark (V. Farsad, A. Gainutdinov, I. Runkel, J. Algebra, 2019)In fact, a f-d (quasi-)Hopf algebra H is factorizable if and only ifRep(H) is non-degenerate.
TheoremLet H be a factorizable semisimple Hopf algebra. AssumedimK(H) = nd where (n,d) = 1 and d is square-free. If Rep(H)is weakly group-theoretical, then H ∼= L× (K[Zd ],Rm).
RemarkFor factorizable s.s quasi-Hopf algebra H with same properties,we have H ∼= A× (Kω[Zd ],R), ω ∈ Z3(Zd ,K∗).
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Motivations Fusion category and braided fusion category Main theorems Applications
Remark (V. Farsad, A. Gainutdinov, I. Runkel, J. Algebra, 2019)In fact, a f-d (quasi-)Hopf algebra H is factorizable if and only ifRep(H) is non-degenerate.
TheoremLet H be a factorizable semisimple Hopf algebra. AssumedimK(H) = nd where (n,d) = 1 and d is square-free. If Rep(H)is weakly group-theoretical, then H ∼= L× (K[Zd ],Rm).
RemarkFor factorizable s.s quasi-Hopf algebra H with same properties,we have H ∼= A× (Kω[Zd ],R), ω ∈ Z3(Zd ,K∗).
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Motivations Fusion category and braided fusion category Main theorems Applications
Remark (V. Farsad, A. Gainutdinov, I. Runkel, J. Algebra, 2019)In fact, a f-d (quasi-)Hopf algebra H is factorizable if and only ifRep(H) is non-degenerate.
TheoremLet H be a factorizable semisimple Hopf algebra. AssumedimK(H) = nd where (n,d) = 1 and d is square-free. If Rep(H)is weakly group-theoretical, then H ∼= L× (K[Zd ],Rm).
RemarkFor factorizable s.s quasi-Hopf algebra H with same properties,we have H ∼= A× (Kω[Zd ],R), ω ∈ Z3(Zd ,K∗).
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Motivations Fusion category and braided fusion category Main theorems Applications
Thank you for your attentions !