Classifying Strictly Weakly Integral Modular Categories of Dimension 16p
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1 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 July 18, 16p / 23
2 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 July 18, 16p / 23
3 Fusion Categories lena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
4 Fusion Categories Abelian C-linear lena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
5 Fusion Categories Abelian C-linear Monoidal (Ob(C),, 1) is a monoid lena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
6 Fusion Categories Abelian C-linear Monoidal (Ob(C),, 1) is a monoid Rigid every object X has left and right duals X lena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
7 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 lena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
8 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 July 18, 16p / 23
9 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 July 18, 16p / 23
10 Modular Categories Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
11 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 July 18, 16p / 23
12 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 Müger 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 July 18, 16p / 23
13 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 Müger 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 Müger center. Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
14 Classifying Modular Categories Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
15 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 July 18, 16p / 23
16 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 July 18, 16p / 23
17 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 July 18, 16p / 23
18 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 July 18, 16p / 23
19 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 = 1 = X X. Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
20 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 July 18, 16p / 23
21 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 July 18, 16p / 23
22 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 July 18, 16p / 23
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 July 18, 16p / 23
24 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 July 18, 16p / 23
25 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 July 18, 16p / 23
26 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 July 18, 16p / 23
27 Fusion Rules For a simple object X, X X = 1 C ad y 1 y X =X y z C ad z >1 N z X,X z Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
28 FPDim(C) = 16p FPdim(X i ) {1, 2, 4, 2, 2 2, p, 2 p, 4 p, 2p, 2 2p} for all simple X i ng {1, 2, p, 2p} E {2, 4} Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
29 FPDim(C) = 16p, GN-Grading dim 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 July 18, 16p / 23
30 FPDim(C) = 16p, GN-Grading dim p 2 p 4 p 2p 2 2p # simples a b c f d h k l m n C int = C E Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
31 FPDim(C) = 16p, GN-Grading dim p 2 p 4 p 2p 2 2p # simples a b c f d h k l m n C int = C E C int = a + 4b + 16c Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
32 FPDim(C) = 16p, GN-Grading dim p 2 p 4 p 2p 2 2p # simples a b c f d h k l m n C int = C E C int = a + 4b + 16c a = C pt = U(C) C pt C int Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
33 FPDim(C) = 16p, GN-Grading dim p 2 p 4 p 2p 2 2p # simples a b c f d h k l m n C int = C E C int = a + 4b + 16c a = C pt = U(C) C pt C int E U(C) Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
34 FPDim(C) = 16p, GN-Grading dim p 2 p 4 p 2p 2 2p # simples a b c f d h k l m n C int = C E C int = a + 4b + 16c a = C pt = U(C) C pt C int 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 July 18, 16p / 23
35 Example case: E = 2, a = 8 dim p 2 p 4 p 2p 2 2p # simples a b c f d h k l m n C g = 2p = a g + 4b g + 16c g 4 2 lena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
36 Example case: E = 2, a = 8 dim p 2 p 4 p 2p 2 2p # simples a b c f d h k l m n C g = 2p = a g + 4b g + 16c g 4 2 a g = 2 in all integral components lena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
37 Example case: E = 2, a = 8 dim p 2 p 4 p 2p 2 2p # simples a b c f d h k l m n C g = 2p = a g + 4b g + 16c g 4 2 a g = 2 in all integral components (C ad ) pt = {1, g} = g g is either modular or symmetric lena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
38 Example case: E = 2, a = 8 dim p 2 p 4 p 2p 2 2p # simples a b c f d h k l m n C g = 2p = a g + 4b g + 16c g 4 2 a g = 2 in all integral components (C ad ) pt = {1, g} = g g is either modular or symmetric If g is symmetric, it is either svec or Rep(Z 2 ) lena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
39 Case i: B = g is modular Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
40 Case i: B = g is modular For a fusion subcategory D C, we denote the relative center by Z C (D) = {X C : C Y,X C X,Y = id X Y Y D} Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
41 Case i: B = g is modular For a fusion subcategory D C, we denote the relative center by Z C (D) = {X C : C Y,X C X,Y = id X Y Y D} If D C are both modular, then Z C (D) is also modular and C = D Z C (D). Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
42 Case i: B = g is modular For a fusion subcategory D C, we denote the relative center by Z C (D) = {X C : C Y,X C X,Y = id X Y Y D} If D C are both modular, then Z C (D) is also modular and C = D Z C (D). C = B Z C (B) Z C (B) = 8p classified by Bruilliard, Plavnik, and Rowell Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
43 Case ii: g =svec Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
44 Case ii: g =svec If C is modular, then C pt = Z C (C ad ). Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
45 Case ii: g =svec If C is modular, then C pt = Z C (C ad ). If D is premodular and g = svec Z C (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 July 18, 16p / 23
46 Case ii: g =svec If C is modular, then C pt = Z C (C ad ). If D is premodular and g = svec Z C (D), then g X X for all simple X D. g = svec C pt = Z C (C ad ) g stabilizes the simple objects of dimension 2 and 4 in C ad, a contradiction Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
47 Case iii: g = Rep(Z 2 ) Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
48 Case iii: g = Rep(Z 2 ) Z 2 -de-equivariantization of C new fusion category C Z2 with FPDim(C Z2 ) = FPDim(C) 2 for each simple X C such that g X = X, there are two simple objects in C Z2 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 C Z2 with dimension FPDim(X ) = FPDim(Y ) Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
49 Case iii: g = Rep(Z 2 ) dim 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 f g 4 p, d g = p fg 4 h g = 2 m g = 1 Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
50 Case iii: g = Rep(Z 2 ) dim 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 f g 4 p, d g = p fg 4 h g = 2 m g = 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 July 18, 16p / 23
51 Case iii: g = Rep(Z 2 ) dim p # simples a b c h Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
52 Case iii: g = Rep(Z 2 ) a = 4 + 2b b = 2c h = 4 dim p # simples a b c h Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
53 Case iii: g = Rep(Z 2 ) a = 4 + 2b b = 2c h = 4 (C Z2 ) int = 4p = 4 + 2b + 8c 2 b dim p # simples a b c h Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
54 Case iii: g = Rep(Z 2 ) a = 4 + 2b b = 2c h = 4 dim p # simples a b c h (C Z2 ) int = 4p = 4 + 2b + 8c 2 b (C Z2 ) pt (C Z2 ) int 4(1 + b p 1 2 ) 4p (b, c) {(0, 2 ), (2p 2, 0)} Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
55 Case iii: g = Rep(Z 2 ) (b, c) {(0, p 1 2 ), (2p 2, 0)} a = 4 + 2b = 4p b = 2c = 0 h = 4 C ad has only two invertibles, so there can be no simple objects of dimension 4 without simple objects of dimension 2. lena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
56 Case iii: g = Rep(Z 2 ) (b, c) {(0, p 1 2 ), (2p 2, 0)} a = 4 + 2b = 4p b = 2c = 0 h = 4 C ad has only two invertibles, so there can be no simple objects of dimension 4 without simple objects of dimension 2. C Z2 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 of invertible objects Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
57 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 ) Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
58 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 pq 4 and p 2 q P. Bruillard, C. Galindo, S.-H. Ng, J. Plavnik, E. C. Rowell, and Z. Wang. On the classification of weakly integral modular categories P. Bruillard, J. Y. Plavnik, and E. C. Rowell. Modular categories of dimension p 3 m with m square-free P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik. Tensor Categories. American Mathematical Society, Elena Amparo (College of William and Mary)Classifying Strictly Weakly Integral Modular Categories of Dimension July 18, 16p / 23
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