classification of fusion categories and factorizable

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

1 Motivations

2 Fusion category and braided fusion category

3 Main theorems

4 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

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.

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.

= ω(g1,g2,g3) :

(Vg1 ⊗ Vg2)⊗ Vg3

= ω(g1,g2,g3) :

(Vg1 ⊗ Vg2)⊗ Vg3

= ω(g1,g2,g3) :

(Vg1 ⊗ Vg2)⊗ Vg3

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)

Definition

Example

FPdim(C) =∑

X∈O(C)

Definition

Example

FPdim(C) =∑

X∈O(C)

Definition

Example

FPdim(C) =∑

X∈O(C)

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 .

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.

ad , m ∈ Z≥1.

Example

ad , m ∈ Z≥1.

Example

ad , m ∈ Z≥1.

Example

ad , m ∈ Z≥1.

Example

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?

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?

Frobenius property

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

(1⊗ 1 + z ⊗ 1 + 1⊗ z − z ⊗ z).

Example

(1⊗ 1 + z ⊗ 1 + 1⊗ z − z ⊗ z).

Example

(1⊗ 1 + z ⊗ 1 + 1⊗ z − z ⊗ z).

Example

(1⊗ 1 + z ⊗ 1 + 1⊗ z − z ⊗ z).

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.

Muger centralizer

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〉).

Example

degenerate.

Example

degenerate.

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.

η(g) = η(g−1), ∀g ∈ G,

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).

Main theorem

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.

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.

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 .

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.

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.

Example

∑i(1⊗ hi)⊗ (hi ⊗ ε).

Rm :=∑

0≤i,j≤n−1

ζ−ijdn

Example

∑i(1⊗ hi)⊗ (hi ⊗ ε).

Rm :=∑

0≤i,j≤n−1

ζ−ijdn

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∗).

Thank you for your attentions !

classification of fusion categories and factorizable

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