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 July 18, 2017 1 / 23
Categories Category C A class of objects Ob( C ) A class of associative morphisms Hom C ( X , Y ) between each pair of objects 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
Fusion Categories Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 3 / 23
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
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
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
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
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
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
Modular Categories Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 4 / 23
Modular Categories Definition A fusion category C is braided if there is a family of natural isomorphisms C X , 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
Modular Categories Definition A fusion category C is braided if there is a family of natural isomorphisms C X , Y : X ⊗ Y − → Y ⊗ X satisfying the hexagon axioms. Definition The M¨ uger center of a braided fusion category C is defined Z 2 ( C ) = { X ∈ C : C Y , X ◦ C X , Y = id X ⊗ Y ∀ Y ∈ C} Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 4 / 23
Modular Categories Definition A fusion category C is braided if there is a family of natural isomorphisms C X , Y : X ⊗ Y − → Y ⊗ X satisfying the hexagon axioms. Definition The M¨ uger center of a braided fusion category C is defined Z 2 ( C ) = { X ∈ C : C Y , X ◦ C X , Y = id X ⊗ 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
Classifying Modular Categories Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 5 / 23
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
Classifying Modular Categories Determine the number of simple objects of each dimension Determine fusion rules X i ⊗ X j = � N X k X i , X j X k N X k X i , X j = [ X i ⊗ X j : X k ] Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 5 / 23
Frobenius-Perron Dimension Definition The Frobenius-Perron Dimension of a simple object X is the largest nonnegative eigenvalue of the matrix N X 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
Frobenius-Perron Dimension Definition The Frobenius-Perron Dimension of a simple object X is the largest nonnegative eigenvalue of the matrix N X of left-multiplication by X . Definition The Frobenius-Perron Dimension of a category C is � FPDim( X i ) 2 summed over all isomorphism classes of simple objects X i ∈ C . Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 6 / 23
Frobenius-Perron Dimension Definition The Frobenius-Perron Dimension of a simple object X is the largest nonnegative eigenvalue of the matrix N X of left-multiplication by X . Definition The Frobenius-Perron Dimension of a category C is � FPDim( X i ) 2 summed over all isomorphism classes of simple objects X i ∈ C . Definition A simple object X is invertible if FPDim( X ) = 1. Equivalently, X ⊗ X ∗ ∼ = X ∗ ⊗ X . = 1 ∼ Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 6 / 23
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
Integral and Weakly Integral Fusion Categories A fusion category C is: pointed if FPDim( X i ) = 1 for all simple X i ∈ C integral if FPDim( X i ) ∈ Z for all simple X i ∈ 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
Integral and Weakly Integral Fusion Categories A fusion category C is: pointed if FPDim( X i ) = 1 for all simple X i ∈ C integral if FPDim( X i ) ∈ Z for all simple X i ∈ C weakly integral if FPDim( C ) ∈ Z In a weakly integral modular category C : FPDim( X i ) 2 � � FPDim( C ) for all simple objects X i ∈ C FPDim( X i ) = √ 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
Grading of a Fusion Category Definition A fusion category C is graded by a group G if: C = ⊕ g ∈ G C g for abelian subcategories C g C g ⊗ C h ⊂ C gh 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
Grading of a Fusion Category Definition A fusion category C is graded by a group G if: C = ⊕ g ∈ G C g for abelian subcategories C g C g ⊗ C h ⊂ C gh for all g , h ∈ G A grading is called faithful if all C g are nonempty. In a faithful grading, all components have dimension FPDim( C ) | G | If a simple object X ∈ C g , then X ∗ ∈ C g − 1 C e ⊃ C ad , 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
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 ) C e = C ad Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 10 / 23
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 ) C e = C ad 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 n g with n e = 1 and FPDim( X ) ∈ √ n g Z for all simple X ∈ C g C e = C int Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension 16p July 18, 2017 10 / 23
Fusion Rules For a simple object X , X ⊗ X ∗ ∼ � � N z = 1 ⊕ y ⊕ X , X ∗ z z ∈C ad C ad ∋ y ≇ 1 | z | > 1 y ⊗ X ∼ = X 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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