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Classifying Strictly Weakly Integral Modular Categories of Dimension 16p Elena Amparo College of William and Mary July 18, 2017 Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p

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Classifying Strictly Weakly Integral Modular Categories

  • f Dimension 16p

Elena Amparo

College of William and Mary

July 18, 2017

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 1 / 23

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Categories

Category C A class of objects Ob(C) A class of associative morphisms HomC(X, Y ) between each pair of

  • bjects X, Y ∈ Ob(C)

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 2 / 23

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Fusion Categories

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 3 / 23

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Fusion Categories

Abelian C-linear

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 3 / 23

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Fusion Categories

Abelian C-linear Monoidal→ (Ob(C), ⊗, 1) is a monoid

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 3 / 23

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Fusion Categories

Abelian C-linear Monoidal→ (Ob(C), ⊗, 1) is a monoid Rigid→ every object X has left and right duals X ∗

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 3 / 23

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Fusion Categories

Abelian C-linear Monoidal→ (Ob(C), ⊗, 1) is a monoid Rigid→ every object X has left and right duals X ∗ Semisimple→ All objects are direct sums of simple objects

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 3 / 23

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Fusion Categories

Abelian C-linear Monoidal→ (Ob(C), ⊗, 1) is a monoid Rigid→ every object X has left and right duals X ∗ Semisimple→ All objects are direct sums of simple objects Finite rank→ Finitely many isomorphism classes of simple objects

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 3 / 23

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Fusion Categories

Abelian C-linear Monoidal→ (Ob(C), ⊗, 1) is a monoid Rigid→ every object X has left and right duals X ∗ Semisimple→ All objects are direct sums of simple objects Finite rank→ Finitely many isomorphism classes of simple objects 1 is simple

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 3 / 23

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Modular Categories

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 4 / 23

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Modular Categories

Definition

A fusion category C is braided if there is a family of natural isomorphisms CX,Y : X ⊗ Y − → Y ⊗ X satisfying the hexagon axioms.

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 4 / 23

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Modular Categories

Definition

A fusion category C is braided if there is a family of natural isomorphisms CX,Y : X ⊗ Y − → Y ⊗ X satisfying the hexagon axioms.

Definition

The M¨ uger center of a braided fusion category C is defined Z2(C) = {X ∈ C : CY ,X ◦ CX,Y = idX⊗Y ∀Y ∈ C}

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 4 / 23

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Modular Categories

Definition

A fusion category C is braided if there is a family of natural isomorphisms CX,Y : X ⊗ Y − → Y ⊗ X satisfying the hexagon axioms.

Definition

The M¨ uger center of a braided fusion category C is defined Z2(C) = {X ∈ C : CY ,X ◦ CX,Y = idX⊗Y ∀Y ∈ C}

Definition

A modular category is a braided, spherical fusion category with trivial M¨ uger center.

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 4 / 23

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Classifying Modular Categories

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 5 / 23

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Classifying Modular Categories

Determine the number of simple objects of each dimension

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 5 / 23

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Classifying Modular Categories

Determine the number of simple objects of each dimension Determine fusion rules Xi ⊗ Xj = NXk

Xi,XjXk

NXk

Xi,Xj = [Xi ⊗ Xj : Xk]

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 5 / 23

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Frobenius-Perron Dimension

Definition

The Frobenius-Perron Dimension of a simple object X is the largest nonnegative eigenvalue of the matrix NX of left-multiplication by X.

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 6 / 23

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Frobenius-Perron Dimension

Definition

The Frobenius-Perron Dimension of a simple object X is the largest nonnegative eigenvalue of the matrix NX of left-multiplication by X.

Definition

The Frobenius-Perron Dimension of a category C is FPDim(Xi)2 summed over all isomorphism classes of simple objects Xi ∈ C.

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 6 / 23

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Frobenius-Perron Dimension

Definition

The Frobenius-Perron Dimension of a simple object X is the largest nonnegative eigenvalue of the matrix NX of left-multiplication by X.

Definition

The Frobenius-Perron Dimension of a category C is FPDim(Xi)2 summed over all isomorphism classes of simple objects Xi ∈ C.

Definition

A simple object X is invertible if FPDim(X) = 1. Equivalently, X ⊗ X ∗ ∼ = 1 ∼ = X ∗ ⊗ X.

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 6 / 23

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Frobenius-Perron Dimension

FPDim(X ⊕ Y ) = FPDim(X) + FPDim(Y ) FPDim(X ⊗ Y ) = FPDim(X)FPDim(Y ) FPDim(X ∗) = FPDim(X)

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 7 / 23

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Integral and Weakly Integral Fusion Categories

A fusion category C is: pointed if FPDim(Xi) = 1 for all simple Xi ∈ C integral if FPDim(Xi) ∈ Z for all simple Xi ∈ C weakly integral if FPDim(C) ∈ Z

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 8 / 23

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Integral and Weakly Integral Fusion Categories

A fusion category C is: pointed if FPDim(Xi) = 1 for all simple Xi ∈ C integral if FPDim(Xi) ∈ Z for all simple Xi ∈ C weakly integral if FPDim(C) ∈ Z In a weakly integral modular category C: FPDim(Xi)2 FPDim(C) for all simple objects Xi ∈ C FPDim(Xi) = √n for some n ∈ Z+

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 8 / 23

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Grading of a Fusion Category

Definition

A fusion category C is graded by a group G if: C = ⊕g∈GCg for abelian subcategories Cg Cg ⊗ Ch ⊂ Cgh for all g, h ∈ G

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 9 / 23

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Grading of a Fusion Category

Definition

A fusion category C is graded by a group G if: C = ⊕g∈GCg for abelian subcategories Cg Cg ⊗ Ch ⊂ Cgh for all g, h ∈ G A grading is called faithful if all Cg are nonempty. In a faithful grading, all components have dimension FPDim(C)

|G|

If a simple object X ∈ Cg, then X ∗ ∈ Cg−1 Ce ⊃ Cad, the smallest fusion subcategory containing X ⊗ X ∗ for all simple X

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 9 / 23

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Grading of a Fusion Category

Universal Grading

Every fusion category is faithfully graded by its universal grading group, U(C) Every faithful grading is a quotient of U(C) In a modular category, U(C) ∼ = G(C) Ce = Cad

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 10 / 23

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Grading of a Fusion Category

Universal Grading

Every fusion category is faithfully graded by its universal grading group, U(C) Every faithful grading is a quotient of U(C) In a modular category, U(C) ∼ = G(C) Ce = Cad

GN-Grading

A weakly integral fusion category is faithfully graded by an elementary abelian 2-group E Simple objects are partitioned by dimension: For each g ∈ E, there is a distinct square-free positive integer ng with ne = 1 and FPDim(X) ∈ √ngZ for all simple X ∈ Cg Ce = Cint

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 10 / 23

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Fusion Rules

For a simple object X, X ⊗ X ∗ ∼ = 1 ⊕

  • Cad∋y≇1

y⊗X∼ =X

y ⊕

  • z∈Cad

|z|>1

Nz

X,X ∗z

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 11 / 23

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FPDim(C) = 16p

FPdim(Xi) ∈ {1, 2, 4, √ 2, 2 √ 2, √p, 2√p, 4√p, √2p, 2√2p} for all simple Xi √ng ∈ {1, √ 2, √p, √2p} |E| ∈ {2, 4}

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 12 / 23

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FPDim(C) = 16p, GN-Grading

dim 1 2 4 √ 2 2 √ 2 √p 2√p 4√p √2p 2√2p # simples a b c f d h k l m n

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 13 / 23

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FPDim(C) = 16p, GN-Grading

dim 1 2 4 √ 2 2 √ 2 √p 2√p 4√p √2p 2√2p # simples a b c f d h k l m n |Cint| = |C|

|E|

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 13 / 23

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FPDim(C) = 16p, GN-Grading

dim 1 2 4 √ 2 2 √ 2 √p 2√p 4√p √2p 2√2p # simples a b c f d h k l m n |Cint| = |C|

|E|

|Cint| = a + 4b + 16c

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 13 / 23

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FPDim(C) = 16p, GN-Grading

dim 1 2 4 √ 2 2 √ 2 √p 2√p 4√p √2p 2√2p # simples a b c f d h k l m n |Cint| = |C|

|E|

|Cint| = a + 4b + 16c a = |Cpt| = |U(C)| |Cpt|

  • |Cint|

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 13 / 23

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FPDim(C) = 16p, GN-Grading

dim 1 2 4 √ 2 2 √ 2 √p 2√p 4√p √2p 2√2p # simples a b c f d h k l m n |Cint| = |C|

|E|

|Cint| = a + 4b + 16c a = |Cpt| = |U(C)| |Cpt|

  • |Cint|

|E|

  • |U(C)|

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 13 / 23

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FPDim(C) = 16p, GN-Grading

dim 1 2 4 √ 2 2 √ 2 √p 2√p 4√p √2p 2√2p # simples a b c f d h k l m n |Cint| = |C|

|E|

|Cint| = a + 4b + 16c a = |Cpt| = |U(C)| |Cpt|

  • |Cint|

|E|

  • |U(C)|

|E| = 2 → a ∈ {4, 4p, 8, 8p} |E| = 4 → a ∈ {4, 4p}

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 13 / 23

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Example case: |E| = 2, a = 8

dim 1 2 4 √ 2 2 √ 2 √p 2√p 4√p √2p 2√2p # simples a b c f d h k l m n |Cg| = 2p = ag + 4bg + 16cg ≡

4 2

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 14 / 23

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Example case: |E| = 2, a = 8

dim 1 2 4 √ 2 2 √ 2 √p 2√p 4√p √2p 2√2p # simples a b c f d h k l m n |Cg| = 2p = ag + 4bg + 16cg ≡

4 2 → ag = 2 in all integral components

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 14 / 23

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Example case: |E| = 2, a = 8

dim 1 2 4 √ 2 2 √ 2 √p 2√p 4√p √2p 2√2p # simples a b c f d h k l m n |Cg| = 2p = ag + 4bg + 16cg ≡

4 2 → ag = 2 in all integral components

(Cad)pt = {1, g} = g → g is either modular or symmetric

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 14 / 23

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Example case: |E| = 2, a = 8

dim 1 2 4 √ 2 2 √ 2 √p 2√p 4√p √2p 2√2p # simples a b c f d h k l m n |Cg| = 2p = ag + 4bg + 16cg ≡

4 2 → ag = 2 in all integral components

(Cad)pt = {1, g} = g → g is either modular or symmetric If g is symmetric, it is either sVec or Rep(Z2)

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 14 / 23

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Case i: B = g is modular

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 15 / 23

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Case i: B = g is modular

For a fusion subcategory D ⊆ C, we denote the relative center by ZC(D) = {X ∈ C : CY ,X ◦ CX,Y = idX⊗Y ∀Y ∈ D}

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 15 / 23

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Case i: B = g is modular

For a fusion subcategory D ⊆ C, we denote the relative center by ZC(D) = {X ∈ C : CY ,X ◦ CX,Y = idX⊗Y ∀Y ∈ D} If D ⊆ C are both modular, then ZC(D) is also modular and C ∼ = D ⊠ ZC(D).

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 15 / 23

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Case i: B = g is modular

For a fusion subcategory D ⊆ C, we denote the relative center by ZC(D) = {X ∈ C : CY ,X ◦ CX,Y = idX⊗Y ∀Y ∈ D} If D ⊆ C are both modular, then ZC(D) is also modular and C ∼ = D ⊠ ZC(D). C ∼ = B ⊠ ZC(B) |ZC(B)| = 8p → classified by Bruilliard, Plavnik, and Rowell

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 15 / 23

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Case ii: g=sVec

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 16 / 23

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Case ii: g=sVec

If C is modular, then Cpt = ZC(Cad).

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 16 / 23

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Case ii: g=sVec

If C is modular, then Cpt = ZC(Cad). If D is premodular and g = sVec ⊂ ZC(D), then g ⊗ X ≇ X for all simple X ∈ D.

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 16 / 23

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Case ii: g=sVec

If C is modular, then Cpt = ZC(Cad). If D is premodular and g = sVec ⊂ ZC(D), then g ⊗ X ≇ X for all simple X ∈ D. g = sVec ⊂ Cpt = ZC(Cad) g stabilizes the simple objects of dimension 2 and 4 in Cad, a contradiction

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 16 / 23

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Case iii: g = Rep(Z2)

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 17 / 23

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Case iii: g = Rep(Z2)

Z2-de-equivariantization of C new fusion category CZ2 with FPDim(CZ2) = FPDim(C)

2

for each simple X ∈ C such that g ⊗ X ∼ = X, there are two simple

  • bjects in CZ2 with dimension FPDim(X)

2

for each pair of simple objects X ≇ Y such that g ⊗ X ∼ = Y (and g ⊗ Y ∼ = X), there is one simple object in CZ2 with dimension FPDim(X) = FPDim(Y )

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 17 / 23

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Case iii: g = Rep(Z2)

dim 1 2 4 √ 2 2 √ 2 √p 2√p 4√p √2p 2√2p # simples a b c f d h k l m n The non-integral components of the universal grading of C have either fg ≡

4 p, dg = p−fg 4

hg = 2 mg = 1

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 18 / 23

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Case iii: g = Rep(Z2)

dim 1 2 4 √ 2 2 √ 2 √p 2√p 4√p √2p 2√2p # simples a b c f d h k l m n The non-integral components of the universal grading of C have either fg ≡

4 p, dg = p−fg 4

hg = 2 mg = 1 Simple objects of dimension √ 2 and √2p are stabilized by g by parity. But

√ 2 2 and √2p 2

cannot be the dimensions of simple objects in a fusion

  • category. So the non-integral component of C must have simple objects of

dimension √p.

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 19 / 23

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Case iii: g = Rep(Z2)

dim 1 2 4 √p # simples a b c h

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 20 / 23

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Case iii: g = Rep(Z2)

dim 1 2 4 √p # simples a b c h a′ = 4 + 2b b′ = 2c h′ = 4

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 20 / 23

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Case iii: g = Rep(Z2)

dim 1 2 4 √p # simples a b c h a′ = 4 + 2b b′ = 2c h′ = 4 |(CZ2)int| = 4p = 4 + 2b + 8c → 2|b

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 20 / 23

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Case iii: g = Rep(Z2)

dim 1 2 4 √p # simples a b c h a′ = 4 + 2b b′ = 2c h′ = 4 |(CZ2)int| = 4p = 4 + 2b + 8c → 2|b |(CZ2)pt|

  • |(CZ2)int| → 4(1 + b

2)|4p → (b, c) ∈ {(0, p−1 2 ), (2p − 2, 0)}

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 20 / 23

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Case iii: g = Rep(Z2)

(b, c) ∈ {(0, p−1

2 ), (2p − 2, 0)}

a′ = 4 + 2b = 4p b′ = 2c = 0 h′ = 4 Cad has only two invertibles, so there can be no simple objects of dimension 4 without simple objects of dimension 2.

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 21 / 23

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Case iii: g = Rep(Z2)

(b, c) ∈ {(0, p−1

2 ), (2p − 2, 0)}

a′ = 4 + 2b = 4p b′ = 2c = 0 h′ = 4 Cad has only two invertibles, so there can be no simple objects of dimension 4 without simple objects of dimension 2. CZ2 is a generalized Tambara-Yamagami category:

Generalized Tambara-Yamagami Category

non-pointed fusion category the tensor product of two non-invertible simple objects is a direct sum

  • f invertible objects

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 21 / 23

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Acknowledgements

Mentor: Dr. Julia Plavnik TAs: Paul Gustafson, Ola Sobieska Collaborators: Katie Lee REU hosted by Texas A&M University and funded by the National Science Foundation (REU grant DMS-1460766)

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 22 / 23

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References

  • P. Bruillard, C. Galindo, S.-M. Hong, Y. Kashina, D. Naidu, S. Natale,
  • J. Y. Plavnik, and E. C. Rowell.

Classification of integral modular categories of Frobenius-Perron dimension pq4 and p2q2. 2013.

  • P. Bruillard, C. Galindo, S.-H. Ng, J. Plavnik, E. C. Rowell, and
  • Z. Wang.

On the classification of weakly integral modular categories. 2014.

  • P. Bruillard, J. Y. Plavnik, and E. C. Rowell.

Modular categories of dimension p3m with m square-free. 2016.

  • P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik.

Tensor Categories. American Mathematical Society, 2010.

Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 23 / 23