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
Motivations Fusion category and braided fusion category Main theorems Applications
1 Motivations
2 Fusion category and braided fusion category
3 Main theorems
4 Applications
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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)
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)
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)
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)
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 .
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 .
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 .
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 .
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.
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.
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.
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.
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.
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?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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).
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).
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).
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).
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.
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.
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.
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.
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.
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.
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.
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.
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〉).
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〉).
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〉).
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.
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.
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.
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.
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.
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.
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).
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).
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).
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).
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).
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).
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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∗).
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∗).
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∗).
Motivations Fusion category and braided fusion category Main theorems Applications
Thank you for your attentions !