Equivalences of graded fusion categories

Edie-Michell, Cain

doi:10.1016/j.jpaa.2020.106652

“…where the C g are full Abelian subcategories and the tensor product bifunctor maps Note that one can have inequivalent extensions which nonetheless are equivalent as tensor categories. This can happen either because the equivalence permutes the gradings, or because the equivalence restricts non-trivially to C; see [Edi21b].…”

Section: Extension Theorymentioning

confidence: 99%

Graded extensions of generalized Haagerup categories

Grossman¹,

Izumi²,

Snyder³

2022

Preprint

0

View full text Add to dashboard Cite

We classify certain Z 2 -graded extensions of generalized Haagerup categories in terms of numerical invariants satisfying polynomial equations. In particular, we construct a number of new examples of fusion categories, including: Z 2 -graded extensions of Z 2n generalized Haagerup categories for all n ≤ 5; Z 2 × Z 2 -graded extensions of the Asaeda-Haagerup categories; and extensions of the Z 2 × Z 2 generalized Haagerup category by its outer automorphism group A 4 . The construction uses endomorphism categories of operator algebras, and in particular, free products of Cuntz algebras with free group C * -algebras.

show abstract

“…We then construct these 2m ′ potential auto-equivalences by the charge conjugation auto-equivalence, which gives us a Z 2 subgroup, and by the canonical Z m ′ -action on C(sl r+1 , k) ad Rep(Z m ′ ) which comes from de-equivariantisation. To obtain the auto-equivalences of C(sl r+1 , k) 0 Rep(Zm) which fix Ω, we appeal to the techniques developed in [15]. These techniques allow us to give an upper bound for EqBr(C(sl r+1 , k) 0 Rep(Zm) ; Ω) in terms of EqBr(C(sl r+1 , k) ad Rep(Z m ′ ) ; Ω) and some cohomogical data.…”

Section: Non-exceptional Auto-equivalences Ofmentioning

confidence: 99%

“…The idea here is to use the fact that C(sl r+1 , k) 0 Rep(Zm) is a Z r+1 mm ′ -graded extension of C(sl r+1 , k) ad Rep(Z m ′ ) . This allows us to apply the results of [15] to classify the auto-equivalences of C(sl r+1 , k) 0 Rep(Zm) extending a given auto-equivalence of C(sl r+1 , k) ad Rep(Z m ′ ) . To convenience the reader, the results of [15] state for a G-graded category ⊕ G C g , the number of auto-equivalences extending F ∈ Eq(C e ) is bounded above by…”

Section: Let Us Define Isomorphisms In Pmentioning

confidence: 99%

“…With this skein theory in hand we can then use standard planar algebra techniques to find the non-exceptional braided auto-equivalence group of the adjoint subcategory. To extend these auto-equivalences to the entire category we use the techniques developed by the author in [15]. These techniques give an upper bound on the number of auto-equivalences which may extend an auto-equivalence on the adjoint subcategory.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Type $II$ quantum subgroups of $\mathfrak{sl}_N$. $I$: Symmetries of local modules

Edie-Michell¹,

Gannon²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

This paper is the first of a pair that aims to classify a large number of the type II quantum subgroups of the categories C(slr+1, k). In this work we classify the braided autoequivalences of the categories of local modules for all known type I quantum subgroups of C(slr+1, k), barring a single unresolved case for an orbifold of C(sl4, 8). We find that the symmetries are all non-exceptional except for four possible cases (up to level-rank duality). These exceptional cases are the orbifolds C(sl2, 16) 0Rep(Z 2 ) , C(sl3, 9) 0 Rep(Z 3 ) , C(sl4, 8) 0 Rep(Z 4 ) , and C(sl5, 5) 0Rep(Z 5 ) . We develop several technical tools in this work. We give a skein theoretic description of the orbifold quantum subgroups of C(slr+1, k). Our methods here are general, and the techniques developed will generalise to give skein theory for any orbifold of a braided tensor category. We also uncover an unexpected connection between quadratic categories and exceptional braided auto-equivalences of the orbifolds. We use this connection to construct two of the four exceptionals, and to reduce the C(sl4, 8) 0Rep(Z 4 ) case to a concrete finite computation. In the sequel to this paper we will use the classified braided auto-equivalences to construct the corresponding type II quantum subgroups of the categories C(slr+1, k). When paired with Gannon's type I classification for r ≤ 6, this will complete the type II classification for these same ranks, excluding the one exception at C(sl4, 8).

show abstract

“…With this skein theory in hand we can then use standard planar algebra techniques to find the non-exceptional braided auto-equivalence group of the adjoint subcategory. To extend these auto-equivalences to the entire category we use the techniques developed by the author in [17]. These techniques give an upper bound on the number of auto-equivalences which may extend an autoequivalence on the adjoint subcategory.…”

Section: Introductionmentioning

confidence: 99%

Type 𝐼𝐼 quantum subgroups of 𝔰𝔩_{𝔑}. ℑ: Symmetries of local modules

Edie-Michell

¹

2023

Comm. Amer. Math. Soc.

Self Cite

1

0

View full text Add to dashboard Cite

This paper is the first of a pair that aims to classify a large number of the type I I II quantum subgroups of the categories C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . In this work we classify the braided auto-equivalences of the categories of local modules for all known type I I quantum subgroups of C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . We find that the symmetries are all non-exceptional except for four cases (up to level-rank duality). These exceptional cases are the orbifolds C ( s l 2 , 16 ) Rep ⁡ ( Z 2 ) 0 \mathcal {C}(\mathfrak {sl}_{2}, 16)^0_{\operatorname {Rep}(\mathbb {Z}_{2})} , C ( s l 3 , 9 ) Rep ⁡ ( Z 3 ) 0 \mathcal {C}(\mathfrak {sl}_{3}, 9)^0_{\operatorname {Rep}(\mathbb {Z}_{3})} , C ( s l 4 , 8 ) Rep ⁡ ( Z 4 ) 0 \mathcal {C}(\mathfrak {sl}_{4}, 8)^0_{\operatorname {Rep}(\mathbb {Z}_{4})} , and C ( s l 5 , 5 ) Rep ⁡ ( Z 5 ) 0 \mathcal {C}(\mathfrak {sl}_{5}, 5)^0_{\operatorname {Rep}(\mathbb {Z}_{5})} . We develop several technical tools in this work. We give a skein theoretic description of the orbifold quantum subgroups of C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . Our methods here are general, and the techniques developed will generalise to give skein theory for any orbifold of a braided tensor category. We also give a formulation of orthogonal level-rank duality in the type D D - D D case, which is used to construct one of the exceptionals. We uncover an unexpected connection between quadratic categories and exceptional braided auto-equivalences of the orbifolds. We use this connection to construct two of the four exceptionals. In the sequel to this paper we will use the classified braided auto-equivalences to construct the corresponding type I I II quantum subgroups of the categories C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . This will essentially finish the type I I II classification for s l n \mathfrak {sl}_n modulo type I I classification. When paired with Gannon’s type I I classification for r ≤ 6 r\leq 6 , our results will complete the type I I II classification for these same ranks. This paper includes an appendix by Terry Gannon, which provides useful results on the dimensions of objects in the categories C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) .

show abstract

Equivalences of graded fusion categories

Cited by 4 publications

References 13 publications

Graded extensions of generalized Haagerup categories

Graded extensions of generalized Haagerup categories

Type $II$ quantum subgroups of $\mathfrak{sl}_N$. $I$: Symmetries of local modules

Type 𝐼𝐼 quantum subgroups of 𝔰𝔩_{𝔑}. ℑ: Symmetries of local modules

Contact Info

Product

Resources

About